Decoding a Message The following cryptogram was encoded

A rectangular array of real numbers that can be used to solve a system of linear equations is called a ________.

A matrix is ________ when the number of rows equals the number of columns.

For a square matrix, the entries are the ________ ________ entries.

A matrix with only one row is called a ________ matrix, and a matrix with only one column is called a ________ matrix.

The matrix derived from a system of linear equations is called the ________ matrix of the system.

The matrix derived from the coefficients of a system of linear equations is called the ________ matrix of the system.

Two matrices are called ________ when one of the matrices can be obtained from the other by a sequence of elementary row operations.

A matrix in row-echelon form is in ________ ________ ________ when every column that has a leading 1 has zeros in every position above and below its leading 1.

In Exercises 916, determine the order of the matrix. 7 0

In Exercises 916, determine the order of the matrix. 5 387

In Exercises 916, determine the order of the matrix. 2 36 3

In Exercises 916, determine the order of the matrix. 3 0 1 7 0 1 15 3 6 0 3 7 2

In Exercises 916, determine the order of the matrix. 33 9 45 20

In Exercises 916, determine the order of the matrix. 7 0 6 5 4 1 3

In Exercises 916, determine the order of the matrix. 1 8 3 6 0 9 1 3 9

In Exercises 916, determine the order of the matrix. 3 4 5 1 1 9 1

In Exercises 1722, write the augmented matrix for the system of linear equations. 4x x 3y  3y  5 12 3

In Exercises 1722, write the augmented matrix for the system of linear equations. 7x 4y 22 5x 9y 15

In Exercises 1722, write the augmented matrix for the system of linear equations. x 10y 5x 3y 2x  y 2z 2 4z 0 6

In Exercises 1722, write the augmented matrix for the system of linear equations. x 7x 3x 8y y   5z 15z 8z    8 38 20  x

In Exercises 1722, write the augmented matrix for the system of linear equations.7x 19x 5y  z 13 8z 10

In Exercises 1722, write the augmented matrix for the system of linear equations. 9x 2y 3z  25y 11z  20 5 

In Exercises 2328, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.) 1 2 2 3 7 4 x

In Exercises 2328, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.) 7 8 5 3 0 2 1

In Exercises 2328, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.) 2 0 6 0 1 3 5 2 0 12 7 2

In Exercises 2328, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.) 4 11 3 5 0 8 1 6 0 18 25 29 2

In Exercises 2328, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.) 9 2 1 3 12 18 7 0 3 5 8 2 0 2 0 0 0 10 4 10 4 11

In Exercises 2328, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.) 614020181710153611 2572321 921

In Exercises 2932, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. 2 3 5 1 1 8 13 3 0 1 39 8 2

In Exercises 2932, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. 3 4 1 3 4 7 3 5 1 0 4 5 3

In Exercises 2932, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. 0 1 4 1 3 5 5 7 1 5 6 3 1 0 0 3 1 7 7 5 27 6 5 27 0

In Exercises 2932, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix. 125254317276 11002963782114 12

In Exercises 3340, fill in the blank(s) using elementary row operations to form a row-equivalent matrix 1 2 4 10 3 5 1 0 4 3 1

In Exercises 3340, fill in the blank(s) using elementary row operations to form a row-equivalent matrix 3 4 6 3 8 6 1 4 3 8 3 61

In Exercises 3340, fill in the blank(s) using elementary row operations to form a row-equivalent matrix 1 5 1 2 1 4 1 0 1 1 1

In Exercises 3340, fill in the blank(s) using elementary row operations to form a row-equivalent matrix 3 18 3 8 12 4 1 1 18 1 8 4 1

In Exercises 3340, fill in the blank(s) using elementary row operations to form a row-equivalent matrix 1 0 0 5 1 0 4 2 1 1 2 7 1 0 0 0 1 0 2 1 2 7

In Exercises 3340, fill in the blank(s) using elementary row operations to form a row-equivalent matrix 1 0 0 0 1 0 6 0 1 1 7 3 1 1 0 0 0 1 0 6 0 1 1 1

In Exercises 3340, fill in the blank(s) using elementary row operations to form a row-equivalent matrix 1 3 2 1 8 1 4 10 12 1 3 6 1 0 0 1 5 3 4 1 1 0 0 1 1 3 4 2 5 1 6 5

In Exercises 3340, fill in the blank(s) using elementary row operations to form a row-equivalent matrix 1 2 4 1 6 8 3 4 3 2 9 1 1 1 2 1 6 3 4 2 9 1 0 1 0 0 2 2 4 7 3 2 1 2 1

In Exercises 41 and 42, (a) perform the row operations to solve the augmented matrix, (b) write and solve the system of linear equations represented by the augmented matrix, and (c) compare the two solution methods. Which do you prefer? 3 6 4 4 22 28 (i) Add to (ii) Add times to (iii) Multiply by (iv) Multiply

In Exercises 41 and 42, (a) perform the row operations to solve the augmented matrix, (b) write and solve the system of linear equations represented by the augmented matrix, and (c) compare the two solution methods. Which do you prefer? 7 3 3 13 5 6 1 1 1 4 4 2 1 3 (i) Add to (ii) Multiply by (iii) Add to (iv) Add times to (v) Add times.

In Exercises 4346, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. 1 0 0 0 1 0 0 1 0 0 5 0

In Exercises 4346, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. 1 0 0 3 0 0 0 1 0 0 8 0

In Exercises 4346, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. 1 0 0 0 1 0 0 0 0 1 1 2

In Exercises 4346, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. 1 0 0 0 1 0 1 0 1 0 2 0

In Exercises 4750, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique. 1 2 3 1 1 6 0 2 7 5 10 14

In Exercises 4750, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique. 1 3 2 2 7 1 1 5 3 3 14 8 1 2

In Exercises 4750, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique. 1 5 6 1 4 8 1 1 18 1 8 0

In Exercises 4750, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique. 1 3 4 3 10 10 0 1 2 7 23 24 1 5

In Exercises 5156, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. 3 1 2 3 0 4 3 4 2

In Exercises 5156, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. 1 5 2 3 15 6 2 9 10

In Exercises 5156, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. 1 1 2 4 2 2 4 8 3 4 4 11 5 9 3 14 1

In Exercises 5156, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.2 4 1 3 3 2 5 8 1 5 2 10 2 8 0 30 1

In Exercises 5156, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. 3 1 5 1 1 1 12 4

In Exercises 5156, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. 5 1 1 5 2 10 4 32

In Exercises 5760, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables and if applicable.) 1 0 2 1 4 3 x,

In Exercises 5760, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables and if applicable.) 1 0 5 1 0 1 1

In Exercises 5760, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables and if applicable.) 1 0 0 1 1 0 2 1 1 4 2 2 1

In Exercises 5760, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables and if applicable.) 1 0 0 2 1 0 2 1 1 1 9 3 1

In Exercises 6164, an augmented matrix that represents a system of linear equations (in variables and if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. 1 0 0 1 3 4 x

In Exercises 6164, an augmented matrix that represents a system of linear equations (in variables and if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. 1 0 0 1 6 10 1

In Exercises 6164, an augmented matrix that represents a system of linear equations (in variables and if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. 1 0 0 0 1 0 0 0 1 4 10 4

In Exercises 6164, an augmented matrix that represents a system of linear equations (in variables and if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. 1 0 0 0 1 0 0 0 1 5 3 0 1

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 2y 7 2x  y 8

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.2x 6y 16 2x 3y  7

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 3x 2y  x 3y  27 13

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x  y  2x 4y  4 34 

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x x   2y 4y y  3z 2z z    28 0 5 

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 3x x x  2y y y   z 2z 4z    15 10 14 

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x  y  3x 4y  4x 8y  22 4 32 

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 2y 0 x  y 6 3x 2y 8

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 3x x 2x x    2y y y y    z 4z 2z z    w 2w w w     0 25 2 6 

In Exercises 6574, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 3x 4x 2x   4y 2y 3y y   3z z 2z 4z   2w 4w w 3w     9 13 4 10  3x x

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. 2x 6y  x 2y  22 9 x

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. 5x 5y  2x 3y  5 7 

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. 8x 4y  5x 2y  7 1

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. x 3y  2x 6y  5 10 

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. x 3x   2y 7y   z 6z   8 26

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. x 2x   y y  4z z   5 9

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. x 3x 2x  y 2y 3z  2z   z  2 5 4 

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. 2x y 2y 7x 5y 3z 24 z 14  6 

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. x 2x 3x   y y 2y   z z z    14 21 19 

In Exercises 7584, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. 2x x x   2y 3y y  z z    2 28 14 

In Exercises 8590, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. 3x 3y 12z  6 x  y  4z  2 2x 5y 20z 10 x 2y  8z  4

In Exercises 8590, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. 2x 10y 2z  x  5y 2z  x  5y  z  3x 15y 3z  6 6 3 9 

In Exercises 8590, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. 2x  y 3x 4y x 5y 5x 2y z 2w   w  2z 6w  z w  6 1 3 3 

In Exercises 8590, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. x 2y 2z  4w  3x 6y 5z 12w  x 3y 3z  2w  6x y z  w  11 30 5 9 

In Exercises 8590, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. x 2x 3x    y 3y 5y z z z  w 0 2w 0 0

In Exercises 8590, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. x 2y x y y z 3w 0  w 0 z 2w 0

In Exercises 9194, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. x 2y  z  y 5z  z  6 16 3  x y 2z  y 3z  z  6 8 3 

In Exercises 9194, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. x 3y 4z  y z  z  11 4 2  x 4y y  3z  z  11 4 2 

In Exercises 9194, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. x 4y 5z  y 7z  z  27 54 8  x 6y  z 15 y 5z 42 z  8

In Exercises 9194, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. x 3y z  y 6z  z  19 18 4  x y 3z  y 2z  z  15 14 4 

In Exercises 95100, use a system of equations to find the quadratic function that satisfies the given conditions. Solve the system using matrices. f11, f2 1, f3 5 f

In Exercises 95100, use a system of equations to find the quadratic function that satisfies the given conditions. Solve the system using matrices. f12, f29, f320

In Exercises 95100, use a system of equations to find the quadratic function that satisfies the given conditions. Solve the system using matrices. f2 15, f17, f1 3 f1

In Exercises 95100, use a system of equations to find the quadratic function that satisfies the given conditions. Solve the system using matrices. f2 3, f1 3, f2 11 f2

In Exercises 95100, use a system of equations to find the quadratic function that satisfies the given conditions. Solve the system using matrices. f18, f213, f320

In Exercises 95100, use a system of equations to find the quadratic function that satisfies the given conditions. Solve the system using matrices. f19, f28, f35

From 2000 through 2011, the numbers of new cases of a waterborne disease in a small city increased in a pattern that was approximately linear (see figure). Find the least squares regression line for the data shown in the figure by solving the following system using matrices. Let represent the year, with corresponding to 2000. Use the result to predict the number of new cases of the waterborne disease in 2014. Is the estimate reasonable? Explain.

Breeding Facility A city zoo borrowed $2,000,000 at simple annual interest to construct a breeding facility. Some of the money was borrowed at 8%, some at 9%, and some at 12%. Use a system of linear equations to determine how much was borrowed at each rate given that the total annual interest was $186,000 and the amount borrowed at 8% was twice the amount borrowed at 12%. Solve the system of linear equations using matrices.

Museum A natural history museum borrowed $2,000,000 at simple annual interest to purchase new exhibits. Some of the money was borrowed at 7%, some at 8.5%, and some at 9.5%. Use a system of linear equations to determine how much was borrowed at each rate given that the total annual interest was $169,750 and the amount borrowed at 8.5% was four times the amount borrowed at 9.5%. Solve the system of linear equations using matrices.

Mathematical Modeling A video of the path of a ball thrown by a baseball player was analyzed with a grid covering the TV screen. The tape was paused three times, and the position of the ball was measured each time. The coordinates obtained are shown in the table. ( and are measured in feet.) (a) Use a system of equations to find the equation of the parabola that passes through the three points. Solve the system using matrices. (b) Use a graphing utility to graph the parabola. (c) Graphically approximate the maximum height of the ball and the point at which the ball struck the ground. (d) Analytically find the maximum height of the ball and the point at which the ball struck the ground. (e) Compare your results from parts (c) and (d).

In Exercises 105 and 106, determine whether the statement is true or false. Justify your answer. 5 1 0 3 2 6 7 0 y is a matrix.

In Exercises 105 and 106, determine whether the statement is true or false. Justify your answer. The method of Gaussian elimination reduces a matrix until a reduced row-echelon form is obtained.

Think About It What is the relationship between the three elementary row operations performed on an augmented matrix and the operations that lead to equivalent systems of equations?

HOW DO YOU SEE IT? Determine whether the matrix below is in row-echelon form, reduced row-echelon form, or neither when it satisfies the given conditions. (a) (b) (c) b 0, c 0 (d) b 0, c 0 b 0, c 0 b 0, c 0 1 c b 1

Two matrices are ________ when their corresponding entries are equal.

When performing matrix operations, real numbers are often referred to as ________.

A matrix consisting entirely of zeros is called a ________ matrix and is denoted by ________.

The matrix consisting of 1s on its main diagonal and 0s elsewhere is called the ________ matrix of order n n

In Exercises 58, find x and y x 7 2 y 4 7 2 22

In Exercises 58, find x and y 5 y x 8 5 12 13 8

In Exercises 58, find x and y 16 3 0 4 13 2 5 15 4 4 6 0 16 3 0 4 13 2 2x 1 15 3y 5 4 3x 0

In Exercises 58, find x and y x 2 1 7 8 2y 2 3 2x y 2 2x 6 1 7 8 18 2 3 8 11

In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A, 3A 2B. A 1 2 1 1, B 2 1 1 8 A

In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A, 3A 2B. A 1 2 2 1  3 4 2 2 A

In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A, 3A 2B. A 8 2 4 1 3 5  B 1 1 1 6 5 10A

In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A, 3A 2B. A 1 0 1 6 3 9, B  2 3 0 4 5 7 A 

In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A, 3A 2B. A  4 1 5 2 1 2 3 1 4 0, B 1 6 0 8 1 2 1 3 0 7 A

In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A, 3A 2B. A  1 3 5 0 4 4 2 4 8 1 0 2 1 6 0  , B  3 2 10 3 0 5 4 9 2 1 1 7 1 4 2 A 

In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A, 3A 2B. A 6 1 0 4 3 0, B  8 4 1 3 A 

In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A, 3A 2B. A B 4623 2 1  , B 

In Exercises 1722, evaluate the expression 5 3 0 67 2 1 1 10 14 8 6 A B

In Exercises 1722, evaluate the expression 6 1 8 00 3 5 1 11 2 7 1 

In Exercises 1722, evaluate the expression 4 0 0 2 1 3 2 3 1 6 2 0

In Exercises 1722, evaluate the expression 1 25 24014 6 18 9 4 

In Exercises 1722, evaluate the expression 3  0 7 3 2 6 8 3 1 2 4 7 4 9 1

In Exercises 1722, evaluate the expression 1 4 2 9 11 1 3 1 6  5 3 0 1 4 137 9 6 5 1 1 3 

In Exercises 2326, use the matrix capabilities of a graphing utility to evaluate the expression. 11 252 1 5 46 3 2 0 2

In Exercises 2326, use the matrix capabilities of a graphing utility to evaluate the expression. 55 14 22 11 19 22 13 20 6

In Exercises 2326, use the matrix capabilities of a graphing utility to evaluate the expression. 1 3.211 1.004 0.055 6.829 4.914 3.889 1.630 5.256 9.768 3.090 8.335 4.251 55 

In Exercises 2326, use the matrix capabilities of a graphing utility to evaluate the expression. 1 10 20 12 15 10 4 1 8  13 7 6 11 0 9  3 3 14 13 8 15 1

In Exercises 2734, solve for in the equation, where X 2A 2B

In Exercises 2734, solve for in the equation, where X 3A 2B

In Exercises 2734, solve for in the equation, where 2X A B

In Exercises 2734, solve for in the equation, where 2X 2A B

In Exercises 2734, solve for in the equation, where 2X 3A B

In Exercises 2734, solve for in the equation, where 3X 4A 2B

In Exercises 2734, solve for in the equation, where 2A 4B 2X 3

In Exercises 2734, solve for in the equation, where 5A 6B 3X 2

In Exercises 3542, if possible, find and state the order of the result. A 2 3 1 1 4 6 , B  0 4 8 1 0 1 0 2 7  AB

In Exercises 3542, if possible, find and state the order of the result. A  0 6 7 1 0 1 2 3 8  , B  2 4 1 1 5 6  A

In Exercises 3542, if possible, find and state the order of the result. A  1 4 0 6 5 3  , B  2 0 3 9 A

In Exercises 3542, if possible, find and state the order of the result. A  1 0 0 0 4 0 0 0 2  , B  3 0 0 0 1 0 0 0 5 

In Exercises 3542, if possible, find and state the order of the result. A  5 0 0 0 8 0 0 0 7  , B  1 5 0 0 0 1 8 0 0 0 1 2 

In Exercises 3542, if possible, find and state the order of the result. A  0 0 0 0 0 0 5 3 4  , B  6 8 0 11 16 0 4 4 0 

In Exercises 3542, if possible, find and state the order of the result. A B 6 216 1

In Exercises 3542, if possible, find and state the order of the result. A 1 6 0 13 3 8 2 17, B  1 4 6 2

In Exercises 4346, use the matrix capabilities of a graphing utility to find if possible. A 7 2 10 5 5 4 4 1 7  , B 2 8 4 2 1 2 3 4 8  AB,

In Exercises 4346, use the matrix capabilities of a graphing utility to find if possible. A  11 14 6 12 10 2 4 12 9  , B  12 5 15 10 12 16 A

In Exercises 4346, use the matrix capabilities of a graphing utility to find if possible. A  3 12 5 8 15 1 6 9 1 8 6 5  , B  3 24 16 8 1 15 10 4 6 14 21 10A

In Exercises 4346, use the matrix capabilities of a graphing utility to find if possible. A  2 21 13 4 5 2 8 6 6  , B  2 7 32 0.5 0 15 14 1.6 B

In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A 1 4 2 2 B 2 1 1 8 A

In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A 6 2 3 4, B  2 2 0 4

In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A 3 1 1 3, B  1 3 3 1 A

In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A 1 1 1 1, B 1 3 3 1 A

In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A B 1127 8 1  ,

In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A 321, B  2 3 0 

In Exercises 5356, evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer  3 0 1 21 2 0 21 2 0 4

In Exercises 5356, evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer 3  6 1 5 2 1 0 0 1 4 3 3 1 

In Exercises 5356, evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer 0 4 2 1 2 2 4 0 1 0 1 2 2 3 0 3 5 3 

In Exercises 5356, evaluate the expression. Use the matrix capabilities of a graphing utility to verify your answer 1 5 7  5 67 18 9  0 4

In Exercises 5764, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordan elimination on to solve for the matrix x x1 2x1   x2 x2 4 0 [A

In Exercises 5764, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordan elimination on to solve for the matrix x 2x1 x1   3x2 4x2 5 10

In Exercises 5764, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordan elimination on to solve for the matrix x 2x1 6x1  3x2 x2 4 36  2

In Exercises 5764, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordan elimination on to solve for the matrix x 4x1 x1  9x2 3x2 13 12 

In Exercises 5764, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordan elimination on to solve for the matrix x x1 2x2 3x3 x1 3x2 x3 2x1 5x2 5x3 9 6 17  4x1

In Exercises 5764, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordan elimination on to solve for the matrix x x1 x1 x1   x2 2x2 x2  3x3 x3 1 1 2  x1

In Exercises 5764, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordan elimination on to solve for the matrix x x1 3x1  5x2 x2 2x2 2x3 x3 5x3 20 8 16  x1

In Exercises 5764, (a) write the system of linear equations as a matrix equation, and (b) use Gauss-Jordan elimination on to solve for the matrix x  x1 x1  x2 3x2 6x2   4x3 5x3 17 11 40 x1

Manufacturing A corporation has four factories, each of which manufactures sport utility vehicles and pickup trucks. The production levels are represented by Factory 1 2 34 Find the production levels when production is increased by 10%.

Vacation Packages A vacation service has identified four resort hotels with a special all-inclusive package. The quoted room rates are for double and family (maximum of four people) occupancy for 5 days and 4 nights. The current rates for the two types of rooms at the four hotels are represented by Hotel Hotel Hotel Hotel wx y z Room rates are guaranteed not to increase by more than 12% by next season. What is the maximum rate per package per hotel?

Agriculture A fruit grower raised two crops, apples and peaches. Each of these crops is shipped to three different outlets. The shipment levels are represented by Outlet 123 The profits per unit are represented by the matrix Compute and interpret the result.

Inventory A company sells five models of computers through three retail outlets. The inventories are represented by The wholesale and retail prices are represented by Compute and interpret the result.

Revenue An electronics manufacturer produces three models of LCD televisions, which are shipped to two warehouses. The shipment levels are represented by Warehouse 1 2 The prices per unit are represented by the matrix Compute and interpret the result.

Labor/Wage Requirements A company that manufactures boats has the following labor-hour and wage requirements. Compute and interpret the result.

Profit At a local dairy mart, the numbers of gallons of skim milk, 2% milk, and whole milk sold over the weekend are represented by Skim 2% Whole milk milk milk The selling prices (in dollars per gallon) and the profits (in dollars per gallon) for the three types of milk sold by the dairy mart are represented by Selling Profit price (a) Compute and interpret the result. (b) Find the dairy marts total profit from milk sales for the weekend.

Voting Preferences The matrix From RD I is called a stochastic matrix. Each entry represents the proportion of the voting population that changes from party to party and represents the proportion that remains loyal to the party from one election to the next. Compute and interpret

Voting Preferences Use a graphing utility to find and for the matrix given in Exercise 72. Can you detect a pattern as is raised to higher powers?

The numbers of calories burned by individuals of different body weights while performing different types of exercises for a one-hour time period are represented by Calories burned 130-lb 155-lb person person (a) A 130-pound person and a 155-pound person played basketball for 2 hours, jumped rope for 15 minutes, and lifted weights for 30 minutes. Organize the times spent exercising in a matrix (b) Compute and interpret the result.

In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer. Two matrices can be added only when they have the same order

In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer. Matrix multiplication is commutative.

In Exercises 7780, use the matrices Show that A B2 A2 2AB B2.

In Exercises 7780, use the matrices Show that A B2 A2 2AB B2.

In Exercises 7780, use the matrices Show that A BA BA2 B2.

In Exercises 7780, use the matrices Show that A B2 A2 AB BA B2.

Think About It If and are real numbers such that and then However, if and are nonzero matrices such that then is not necessarily equal to Illustrate this using the following matrices. A ,  0 0 1 1, B 1 1 0 0A ,  C  2 2 3 3 B 1

Think About It If and are real numbers such that then or However, if and are matrices such that it is not necessarily true that or Illustrate this using the following matrices. A 3 4 3 4, B 1 1 1 1 A

Conjecture Let and be unequal diagonal matrices of the same order. (A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero.) Determine the products for several pairs of such matrices. Make a conjecture about a quick rule for such products

Matrices with Complex Entries Let and let and (a) Find and Identify any similarities with and (b) Find and identify

Finding Matrices Find two matrices and such that AB BA.

HOW DO YOU SEE IT? A corporation has three factories, each of which manufactures acoustic guitars and electric guitars. The production levels are represented by Factory AB C (a) Interpret the value of (b) How could you find the production levels when production is increased by 20%? (c) Each acoustic guitar sells for $80 and each electric guitar sells for $120. How could you use matrices to find the total sales value of the guitars produced at each factory?

In a ________ matrix, the number of rows equals the number of columns

If there exists an matrix such that then is called the ________ of

3. If a matrix has an inverse, then it is called invertible or ________; if it does not have an inverse, then it is called ________.

If is an invertible matrix, then the system of linear equations represented by has a unique solution given by ________.

In Exercises 512, show that is the inverse of A A 2 5 1 3, B 3 5 1 2 A 2

In Exercises 512, show that is the inverse of A A 1 1 1 2, B B 2 1 1 1 A

In Exercises 512, show that is the inverse of A A 1 3 2 4, B 1 2 4 3 2 1 A 1

In Exercises 512, show that is the inverse of A A 1 2 1 3, B B 1 53 2 1 1 A

In Exercises 512, show that is the inverse of A A 2 1 0 17 11 3 11 7 2 , B 1 2 3 1 4 6 2 3 5

In Exercises 512, show that is the inverse of A A 4 1 0 1 2 1 5 4 1 , B 1 4 2 1 1 4 4 4 6 11 7 A

In Exercises 512, show that is the inverse of A A 2 3 1 3 0 0 1 1 2 0 2 1 1 1 1 0 , B 1 3 1 2 1 3 3 9 0 6 2 7 1 6 2 10 1 6 A 2

In Exercises 512, show that is the inverse of A A 1 1 1 0 1 1 1 1 0 1 2 1 1 0 0 1 , B 1 3 3 3 0 3 1 1 1 2 1 2 1 1 3 3 0 0 A

In Exercises 1324, find the inverse of the matrix (if it exists). 2 0 0 3

In Exercises 1324, find the inverse of the matrix (if it exists). 1 3 2 7

In Exercises 1324, find the inverse of the matrix (if it exists). 1 2 2 3

In Exercises 1324, find the inverse of the matrix (if it exists). 7 4 33 19 1

In Exercises 1324, find the inverse of the matrix (if it exists). 3 4 1 2

In Exercises 1324, find the inverse of the matrix (if it exists). 4 3 1 1 3

In Exercises 1324, find the inverse of the matrix (if it exists). 1 3 3 1 5 6 1 4 5

In Exercises 1324, find the inverse of the matrix (if it exists). 3 1 2 7 4 2 9 7 1

In Exercises 1324, find the inverse of the matrix (if it exists). 5 2 1 0 0 5 0 0 7

In Exercises 1324, find the inverse of the matrix (if it exists). 1 3 2 0 0 5 0 0 5

In Exercises 1324, find the inverse of the matrix (if it exists). 8 0 0 0 0 1 0 0 0 0 4 0 0 0 0 5

In Exercises 1324, find the inverse of the matrix (if it exists). 0 0 0 3 2 0 0 2 4 2 0 0 6 1 5 8

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 1 3 5 2 7 7 1 10 15 1

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 10 5 3 5 1 2 7 4 2 1

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 1 3 2 1 1 0 2 0 3

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 3 2 4 2 2 4 2 2 3

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 1 2 1 0 3 4 0 1 1 4 3 2 1 2

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 6 0 1 1 3 2 3 1 2 11 6 2 5

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 0.1 0.3 0.5 0.2 0.2 0.4 0.3 0.2 0.4

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 0.6 0.7 1 0 1 0 0.3 0.2 0.9 0.

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 1 0 2 0 0 2 0 1 1 0 1 0 0 1 0 1

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 3 2 1 2 5 5 4 1 2 2 4 2 3 5 11 1 0 2 0

In Exercises 3540, use the formula on page 527 to find the inverse of the matrix (if it exists). 2 1 3 5 2

In Exercises 3540, use the formula on page 527 to find the inverse of the matrix (if it exists). 1 3 2 2 2

In Exercises 3540, use the formula on page 527 to find the inverse of the matrix (if it exists). 4 2 6 3

In Exercises 3540, use the formula on page 527 to find the inverse of the matrix (if it exists). 12 5 3 2 4

In Exercises 3540, use the formula on page 527 to find the inverse of the matrix (if it exists). 7 2 1 5 3 4 4 5

In Exercises 3540, use the formula on page 527 to find the inverse of the matrix (if it exists). 1 4 5 3 9 4 8 9

In Exercises 4144, use the inverse matrix found in Exercise 15 to solve the system of linear equations x 2y 5 2x 3y 10

In Exercises 4144, use the inverse matrix found in Exercise 15 to solve the system of linear equations x 2y 0 2x 3y 3  x

In Exercises 4144, use the inverse matrix found in Exercise 15 to solve the system of linear equations x 2y 4 2x 3y 2 

In Exercises 4144, use the inverse matrix found in Exercise 15 to solve the system of linear equations x 2y 2x 3y 1 2  x

In Exercises 45 and 46, use the inverse matrix found in Exercise 19 to solve the system of linear equations. x  y  z 0 3x 5y 4z 5 3x 6y 5z 2 

In Exercises 45 and 46, use the inverse matrix found in Exercise 19 to solve the system of linear equations. x  y  z 3x 5y 4z 3x 6y 5z 1 2 0 

In Exercises 47 and 48, use the inverse matrix found in Exercise 34 to solve the system of linear equations. x1 2x2 x3 2x4 3x1 5x2 2x3 3x4 2x1 5x2 2x3 5x4 x1 4x2 4x3 11x4 0 1 1 2  x  y  z

In Exercises 47 and 48, use the inverse matrix found in Exercise 34 to solve the system of linear equations. x1 2x2 x3 2x4 3x1 5x2 2x3 3x4 2x1 5x2 2x3 5x4 x1 4x2 4x3 11x4 1 2 0 3  x1 2x2 x3

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations. 3x 4y 5x 3y 2 4 

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations.18x 12y 13 30x 24y 23 

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations. 0.4x  2x 0.8y 4y 1.6 5 

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations. 0.2x x  0.6y 1.4y 2.4 8.8  0

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations. 1 4 x 3 8 y 2 3 2 x 3 4 y 12  0

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations.  5 6 x 4 3 x y 20 7 2 y 51  1

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations. 4x y  z 2x 2y 3z 5x 2y 6z 5 10 1  5

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations. 4x 2y 3z 2x 2y 5z 8x 5y 2z 2 16 4  4x

In Exercises 5760, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations  5x 3y 2z 2x 2y 3z x 7y 8z 2 3 4  4x

In Exercises 5760, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations 2x 3y  5z 4 3x 5y  9z 7 5x 9y 17z 13 

In Exercises 5760, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations 3x 2y  z 4x  y 3z x 5y  z 29 37 24  2x 

In Exercises 5760, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations 8x 7y 10z 12x 3y 5z 15x 9y  2z 151 86 187  3x

In Exercises 6164, you invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7% on A bonds, and 9% on B bonds. You invest twice as much in B bonds as in A bonds. Let and represent the amounts invested in AAA, A, and B bonds, respectively. Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. 61. $10,000 $705

In Exercises 6164, you invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7% on A bonds, and 9% on B bonds. You invest twice as much in B bonds as in A bonds. Let and represent the amounts invested in AAA, A, and B bonds, respectively. Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. 62. $10,000 $760

In Exercises 6164, you invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7% on A bonds, and 9% on B bonds. You invest twice as much in B bonds as in A bonds. Let and represent the amounts invested in AAA, A, and B bonds, respectively. Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. 63. $12,000 $835

In Exercises 6164, you invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. The average yields are 6.5% on AAA bonds, 7% on A bonds, and 9% on B bonds. You invest twice as much in B bonds as in A bonds. Let and represent the amounts invested in AAA, A, and B bonds, respectively. Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. 64. $500,000 $38,000

In Exercises 6568, consider the circuit shown in the figure. The currents and in amperes, are the solution of the system of linear equations. where and are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. E1 15 volts, E2 17volts

In Exercises 6568, consider the circuit shown in the figure. The currents and in amperes, are the solution of the system of linear equations. where and are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. E1 10 volts, E2 10volts

In Exercises 6568, consider the circuit shown in the figure. The currents and in amperes, are the solution of the system of linear equations. where and are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. E1 28 volts, E2 21 volts

In Exercises 6568, consider the circuit shown in the figure. The currents and in amperes, are the solution of the system of linear equations. where and are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. E1 24 volts, E volts 2 23volts

In Exercises 69 and 70, find the numbers of bags of potting soil that a company can produce for seedlings, general potting, and hardwood plants with the given amounts of raw materials. The raw materials used in one bag of each type of potting soil are shown below. 500 units of sand 500 units of loam400 units of peat moss

In Exercises 69 and 70, find the numbers of bags of potting soil that a company can produce for seedlings, general potting, and hardwood plants with the given amounts of raw materials. The raw materials used in one bag of each type of potting soil are shown below. 500 units of sand 750 units of loam 450 units of peat moss

Floral Design A florist is creating 10 centerpieces for the tables at a wedding reception. Roses cost $2.50 each, lilies cost $4 each, and irises cost $2 each. The customer has a budget of $300 allocated for the centerpieces and wants each centerpiece to contain 12 flowers, with twice as many roses as the number of irises and lilies combined. (a) Write a system of linear equations that represents the situation. (b) Write a matrix equation that corresponds to your system. (c) Solve your system of linear equations using an inverse matrix. Find the number of flowers of each type that the florist can use to create the 10 centerpieces.

International Travel The table shows the numbers of international travelers (in thousands) to the United States from South America from 2008 through 2010. (Source: U.S. Department of Commerce) (a) The data can be modeled by the quadratic function Create a system of linear equations for the data. Let represent the year, with corresponding to 2008. (b) Use the matrix capabilities of a graphing utility to find the inverse matrix to solve the system from part (a) and find the least squares regression parabola (c) Use the graphing utility to graph the parabola with the data. (d) Do you believe the model is a reasonable predictor of future numbers of travelers? Explain.

In Exercises 73 and 74, determine whether the statement is true or false. Justify your answer. Multiplication of an invertible matrix and its inverse is commutative.

In Exercises 73 and 74, determine whether the statement is true or false. Justify your answer. When the product of two square matrices is the identity matrix, the matrices are inverses of one another.

Writing Explain how to determine whether the inverse of a matrix exists. If so, explain how to find the inverse

Writing Explain in your own words how to write a system of three linear equations in three variables as a matrix equation, as well as how to solve the system using an inverse matrix

Conjecture Consider matrices of the form (a) Write a matrix and a matrix in the form of Find the inverse of each. (b) Use the result of part (a) to make a conjecture about the inverses of matrices in the form of A

HOW DO YOU SEE IT? Let be the matrix Use the determinant of to state the conditions for which (a) exists and (b) A A 1 A. 1 A A x 0 y z. A 2 2 Sand

Both and represent the ________ of the matrix A

The ________ of the entry is the determinant of the matrix obtained by deleting the th row and th column of the square matrix A

The ________ of the entry of the square matrix is given by Mij a A . Cij ij

The method of finding the determinant of a matrix of order or greater is ________ by ________.

In Exercises 522, find the determinant of the matrix. 4

In Exercises 522, find the determinant of the matrix. 10

In Exercises 522, find the determinant of the matrix. 8 2 4 3

In Exercises 522, find the determinant of the matrix. 9 6 0 2 8

In Exercises 522, find the determinant of the matrix. 6 5 2 3

In Exercises 522, find the determinant of the matrix. 3 4 3 8 

In Exercises 522, find the determinant of the matrix. 7 3 0 0

In Exercises 522, find the determinant of the matrix. 4 0 3 0

In Exercises 522, find the determinant of the matrix. 2 0 6 3

In Exercises 522, find the determinant of the matrix. 2 6 3 9 

In Exercises 522, find the determinant of the matrix. 3 6 2 1

In Exercises 522, find the determinant of the matrix. 4 2 7 5

In Exercises 522, find the determinant of the matrix. 2 3 7 1

In Exercises 522, find the determinant of the matrix. 2 4 5 1 

In Exercises 522, find the determinant of the matrix. 7 1 2 6 3

In Exercises 522, find the determinant of the matrix. 0 3 6 2

In Exercises 522, find the determinant of the matrix. 1 2 6 1 3 1 3

In Exercises 522, find the determinant of the matrix. 2 3 1 4 3 1

In Exercises 2330, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 3 2 4 1

In Exercises 2330, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 5 7 9 16

In Exercises 2330, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 19 43 20 56

In Exercises 2330, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 101 253 197 172

In Exercises 2330, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 0.3 0.2 0.2 0.2

In Exercises 2330, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 0.1 0.3 0.2 0.2

In Exercises 2330, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 0.9 0.1 0.7 0.3

In Exercises 2330, use the matrix capabilities of a graphing utility to find the determinant of the matrix. 0.1 7.5 0.1 6.2

In Exercises 3138, find all the (a) minors and (b) cofactors of the matrix. 4 3 5 6

In Exercises 3138, find all the (a) minors and (b) cofactors of the matrix.  0 3 10 4

In Exercises 3138, find all the (a) minors and (b) cofactors of the matrix. 3 2 1 4

In Exercises 3138, find all the (a) minors and (b) cofactors of the matrix. 6 7 5 2

In Exercises 3138, find all the (a) minors and (b) cofactors of the matrix. 4 3 1 0 2 1 2 1 1

In Exercises 3138, find all the (a) minors and (b) cofactors of the matrix. 1 3 4 1 2 6 0 5 4 4

In Exercises 3138, find all the (a) minors and (b) cofactors of the matrix. 4 7 1 6 2 0 3 8 5

In Exercises 3138, find all the (a) minors and (b) cofactors of the matrix. 2 7 6 9 6 7 4 0 6 

In Exercises 3946, find the determinant of the matrix. Expand by cofactors using the indicated row or column. 3 4 2 2 5 3 1 6 1 (a) Row 1 ((b) Column 2

In Exercises 3946, find the determinant of the matrix. Expand by cofactors using the indicated row or column. 3 6 4 4 3 7 2 1 8  (a) Row 2 b) Column 3

In Exercises 3946, find the determinant of the matrix. Expand by cofactors using the indicated row or column. 5 0 1 0 12 6 3 4 3 (a) Row 2 (b) Column 2

In Exercises 3946, find the determinant of the matrix. Expand by cofactors using the indicated row or column. 10 30 0 5 0 10 5 10 1  (a) Row 3 (b) Column 1

In Exercises 3946, find the determinant of the matrix. Expand by cofactors using the indicated row or column. 2 3 8 6 4 0 5 0 7 0 10 5 1 0 5 0 (a) Row 2 (b) Column 4

In Exercises 3946, find the determinant of the matrix. Expand by cofactors using the indicated row or column. 7 6 1 3 0 0 2 0 0 1 3 1 6 2 2 4 2 (a) Row 4 (b) Column 2

In Exercises 3946, find the determinant of the matrix. Expand by cofactors using the indicated row or column. 6 4 1 8 0 13 0 6 3 6 7 0 5 8 4 2  (a) Row 2 (b) Column 2

In Exercises 3946, find the determinant of the matrix. Expand by cofactors using the indicated row or column. 10 4 0 1 8 0 3 0 3 5 2 3 7 6 7 2  (a) Row 3 (b) Column 1

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 2 4 4 1 2 2 0 1 1

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 2 1 0 2 1 1 3 0 4

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 6 0 4 3 0 6 7 0 3

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 3 2 1 1 0 2 0 3

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 1 0 0 8 3 0 3 6 3 

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 1 1 4 0 1 11 0 0 5 

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 1 3 1 4 2 4 2 0 3

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest.  2 1 1 1 4 0 3 4 2

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 2 0 0 4 3 0 6 1 5

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 3 7 1 0 11 2 0 0 2

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 2 2 1 3 6 7 5 7 6 3 0 0 2 6 1 7

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 3 2 1 0 6 0 1 3 5 6 2 1 4 0 2 1 2

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 5 4 0 0 3 6 2 1 0 4 3 2 6 12 4 2

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 1 5 0 3 4 6 0 2 3 2 0 1 2 1 0 5 

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 3 2 1 6 3 2 0 0 0 0 4 1 0 2 5 1 3 4 1 1 5 2 0 0 0 

In Exercises 4762, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 5 0 0 0 0 2 1 0 0 0 0 4 2 3 0 0 3 6 4 0 2 2 3 1 2

In Exercises 6366, use the matrix capabilities of a graphing utility to evaluate the determinant. 3 0 8 8 5 1 7 4 6

In Exercises 6366, use the matrix capabilities of a graphing utility to evaluate the determinant. 5 9 8 8 7 7 0 4 13

In Exercises 6366, use the matrix capabilities of a graphing utility to evaluate the determinant. 1 2 2 0 1 6 0 2 8 0 2 8 4 4 6 0

In Exercises 6366, use the matrix capabilities of a graphing utility to evaluate the determinant. 0 8 4 7 3 1 6 0 8 1 0 0 2 6 9 141

In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 1 0 0 3, B  2 0 0 1 A

In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 2 4 1 2, B B  1 0 2 1 A

In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 4 3 0 2, B  1 2 1 2 A

In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 5 3 4 1, B  0 1 6 2 A

In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 0 3 0 1 2 4 2 1 1 B  3 1 3 2 1 1 0 2 1A

In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 3 1 2 2 3 0 0 4 1 , B  3 0 2 0 2 1 1 1 1 A

In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A  1 1 0 2 0 1 1 1 0 B  1 0 0 0 2 0 0 0 3A

In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A  2 1 3 0 1 1 1 2 0 B  2 0 3 1 1 2 4 3 1A

In Exercises 7580, evaluate the determinant(s) to verify the equation. w y x z y w z x

In Exercises 7580, evaluate the determinant(s) to verify the equation. w y x z y w z x

In Exercises 7580, evaluate the determinant(s) to verify the equation. w y x z w y x cw z cy

In Exercises 7580, evaluate the determinant(s) to verify the equation. w cw x cx0

In Exercises 7580, evaluate the determinant(s) to verify the equation. 1 1 x y z x 2 y 2 z 2 y xz xz y w cw

In Exercises 7580, evaluate the determinant(s) to verify the equation. a b a a a a b a a a a bb23a b

In Exercises 8188, solve for x x 1 2 x2

In Exercises 8188, solve for x x 1 4 x20 

In Exercises 8188, solve for x x 2 1 x 2 1 

In Exercises 8188, solve for x x 1 1 2 x4 

In Exercises 8188, solve for x x 1 3 2 x 20 

In Exercises 8188, solve for x x 2 3 1 x0 

In Exercises 8188, solve for x x 3 1 2 x 20

In Exercises 8188, solve for x x 4 7 2 x 50 

In Exercises 8994, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. 4u 1 1 2v

In Exercises 8994, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.  3x 2 1 3y 2 1

In Exercises 8994, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. e2x 2e2x e3x 3e3

In Exercises 8994, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. ex ex xex 1 xex  e2

In Exercises 8994, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. x 1 ln x 1 x

In Exercises 8994, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. x 1 x ln x 1 ln x

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, then the determinant will always be zero.

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer. If two columns of a square matrix are the same, then the determinant of the matrix will be zero.

Providing a Counterexample Find square matrices and to demonstrate that A BAB.

Conjecture Consider square matrices in which the entries are consecutive integers. An example of such a matrix is (a) Use a graphing utility to evaluate the determinants of four matrices of this type. Make a conjecture based on the results. (b) Verify your conjecture.

Writing Write a brief paragraph explaining the difference between a square matrix and its determinant

Think About It Let be a matrix such that Is it possible to find Explain.

In Exercises 101103, a property of determinants is given ( and are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If is obtained from by interchanging two rows of or interchanging two columns of then B AA A, . B

In Exercises 101103, a property of determinants is given ( and are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. f is obtained from by adding a multiple of a row of to another row of or by adding a multiple of a column of to another column of then BAA A, .

In Exercises 101103, a property of determinants is given ( and are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If is obtained from by multiplying a row by a nonzero constant or by multiplying a column by a nonzero constant then BcAc, .

HOW DO YOU SEE IT? Explain why the determinant of the matrix is equal to zero. (a) (b) (c) (d)  4 2 4 6 4 2 4 1 5 3 5 3 7 1 7 3  2 1 0 4 2 0 5 3 0  3 6 5 2 4 7 1 2 9 3 1 0 1 4 3 5 3 2 1 7 1 7 2 1 2  7.4 The D

Conjecture A diagonal matrix is a square matrix with all zero entries above and below its main diagonal. Evaluate the determinant of each diagonal matrix. Make a conjecture based on your results. (a) (b) (c)  2 0 0 0 0 2 0 0 0 0 1 0 0 0 0 3  1 0 0 0 5 0 0 0 2  7 0 0 4

The method of using determinants to solve a system of linear equations is called ________ ________.

Three points are ________ when the points lie on the same line.

The area of a triangle with vertices and is given by ________.

A message written according to a secret code is called a ________.

To encode a message, choose an invertible matrix and multiply the ________ row matrices by (on the right) to obtain ________ row matrices.

If a message is encoded using an invertible matrix then the message can be decoded by multiplying the coded row matrices by ________ (on the right).

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 7x 3x  11y 9y 1 9 A l

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 4x 6x  3y 9y 10 12 7x

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 3x 2y 6x 4y 2 4 

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 6x 5y 13x 3y 17 76  3

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 0.4x 0.8y 1.6 0.2x 0.3y 2.2

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 2.4x 1.3y 4.6x 0.5y 14.63 11.51  0.

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 4x y  z 2x 2y 3z 5x 2y 6z 5 10 1  2

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 4x 2y 3z 2x 2y 5z 8x 5y 2z 2 16 4  4x

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations x 2y 3z 2x  y z 3x 3y 2z 3 6 11  4x

In Exercises 716, use Cramers Rule to solve (if possible) the system of equations 5x 4y  z x 2y 2z 3x  y  z 14 10 1 x 2y

In Exercises 1728, use a determinant to find the area with the given vertices.

In Exercises 1728, use a determinant to find the area with the given vertices.

In Exercises 1728, use a determinant to find the area with the given vertices.

In Exercises 1728, use a determinant to find the area with the given vertices.

In Exercises 1728, use a determinant to find the area with the given vertices. 0, 4, 5, 6, 10, 6, 11 2, 4, 3,  5 2, 0

In Exercises 1728, use a determinant to find the area with the given vertices. 4, 5, 6, 10, 6, 11

In Exercises 1728, use a determinant to find the area with the given vertices. 2, 4, 2, 3, 1, 5

In Exercises 1728, use a determinant to find the area with the given vertices. 0, 2, 1, 4, 3, 5

In Exercises 1728, use a determinant to find the area with the given vertices. 3, 5, 2, 6, 3, 5

In Exercises 1728, use a determinant to find the area with the given vertices. 2, 4, 1, 5, 3, 2

In Exercises 1728, use a determinant to find the area with the given vertices. 4, 2, 0, 7 2, 3, 1 2

In Exercises 1728, use a determinant to find the area with the given vertices. 9 2, 0, 2, 6, 0, 3 2

In Exercises 29 and 30, find a value of such that the triangle with the given vertices has an area of 4 square units. 5, 1, 0, 2, 2, y4

In Exercises 29 and 30, find a value of such that the triangle with the given vertices has an area of 4 square units. 4, 2, 3, 5, 1, y

In Exercises 31 and 32, find a value of such that the triangle with the given vertices has an area of 6 square units. 2, 3, 1, 1, 8, y y

In Exercises 31 and 32, find a value of such that the triangle with the given vertices has an area of 6 square units. 1, 0, 5, 3, 3, y

A large region of forest has been infested with gypsy moths. The region is roughly triangular, as shown in the figure. From the northernmost vertex of the region, the distances to the other vertices are 25 miles south and 10 miles east (for vertex ), and 20 miles south and 28 miles east (for vertex ). Use a graphing utility to approximate the number of square miles in this region.

Botany A botanist is studying the plants growing in a triangular tract of land, as shown in the figure. To estimate the number of square feet in the tract, the botanist starts at one vertex, walks 65 feet east and 50 feet north to the second vertex, and then walks 85 feet west and 30 feet north to the third vertex. Use a graphing utility to determine how many square feet there are in the tract of land.

In Exercises 3540, use a determinant to determine whether the points are collinear. 3, 1, 0, 3, 12, 5 6

In Exercises 3540, use a determinant to determine whether the points are collinear. 3, 5, 6, 1, 4, 2

In Exercises 3540, use a determinant to determine whether the points are collinear. 2, 1 2, 4, 4, 6, 3 

In Exercises 3540, use a determinant to determine whether the points are collinear. 0, 1 2, 2, 1, 4, 7

In Exercises 3540, use a determinant to determine whether the points are collinear. 0, 2, 1, 2.4, 1, 1.6

In Exercises 3540, use a determinant to determine whether the points are collinear. 2, 3, 3, 3.5, 1, 2

In Exercises 41 and 42, find such that the points are collinear. 2, 5, 4, y, 5, 2

In Exercises 41 and 42, find such that the points are collinear. 6, 2, 5, y, 3, 5

In Exercises 4348, use a determinant to find an equation of the line passing through the points. 0, 0, 5, 3

In Exercises 4348, use a determinant to find an equation of the line passing through the points. 0, 0, 2, 2

In Exercises 4348, use a determinant to find an equation of the line passing through the points. 4, 3, 2, 1

In Exercises 4348, use a determinant to find an equation of the line passing through the points. 10, 7, 2, 7

In Exercises 4348, use a determinant to find an equation of the line passing through the points. , 4, 6, 121 2, 3, 5 2, 1

In Exercises 4348, use a determinant to find an equation of the line passing through the points. 2 3 , 4, 6, 12

In Exercises 49 and 50, (a) write the uncoded row matrices for the message, and then (b) encode the message using the encoding matrix. COME HOME SOON 1 3 2 5

In Exercises 49 and 50, (a) write the uncoded row matrices for the message, and then (b) encode the message using the encoding matrix. HELP IS ON THE WAY 2 1 3 1

In Exercises 51 and 52, (a) write the uncoded row matrices for the message, and then (b) encode the message using the encoding matrix. CALL ME TOMORROW 1 1 6 1 0 2 0 1 3 

In Exercises 51 and 52, (a) write the uncoded row matrices for the message, and then (b) encode the message using the encoding matrix. PLEASE SEND MONEY 4 3 3 2 3 2 1 1 1 

In Exercises 5356, write a cryptogram for the message using the matrix LANDING SUCCESSFUL

In Exercises 5356, write a cryptogram for the message using the matrix ICEBERG DEAD AHEAD

In Exercises 5356, write a cryptogram for the message using the matrix HAPPY BIRTHDAY

In Exercises 5356, write a cryptogram for the message using the matrix OPERATION OVERLOAD

In Exercises 5760, use to decode the cryptogram. A 1 3 2 5 11 21 64 112 25 50 29 53 23 46 40 75 55 92

In Exercises 5760, use to decode the cryptogram. A 2 3 3 4 A 85 120 6 8 10 15 84 117 42 56 90 125 60 80 30 45 19 26

In Exercises 5760, use to decode the cryptogram. A 1 1 6 1 0 2 0 1 3  A 1 9 19 19 9 19 A 1 25 41 21 31 9

In Exercises 5760, use to decode the cryptogram. A 3 0 4 4 2 5 2 1 3 8 112 83 19 13 72 61 95 71 20 21 38 35 36 42 32

In Exercises 61 and 62, decode the cryptogram by using the inverse of the matrix in Exercises 5356. 20 17 1 62 143 181

In Exercises 61 and 62, decode the cryptogram by using the inverse of the matrix in Exercises 5356. 13 61 112 106 11 24 29 65 144 172

Decoding a Message The following cryptogram was encoded with a matrix. 8 21 5 10 5 25 5 19 6 20 40 1 16 The last word of the message is _RON. What is the message?

Decoding a Message The following cryptogram was encoded with a matrix. 5 2 25 11 32 14 8 38 19 37 16 The last word of the message is _SUE. What is the message?

Circuit Analysis Consider the circuit in the figure. The currents and in amperes are given by the solution of the system of linear equations. Use Cramers Rule to find the three currents.

. Pulley System A system of pulleys that is assumed frictionless and without mass is loaded with 192-pound and 64-pound weights (see figure). The tensions and in the ropes and the acceleration of the 64-pound weight are found by solving the system where and are measured in pounds and is in feet per second squared. Use Cramers Rule to find and

In Exercises 6770, determine whether the statement is true or false. Justify your answer. In Cramers Rule, the numerator is the determinant of the coefficient matrix.

In Exercises 6770, determine whether the statement is true or false. Justify your answer. You cannot use Cramers Rule to solve a system of linear equations when the determinant of the coefficient matrix is zero.

In Exercises 6770, determine whether the statement is true or false. Justify your answer. In a system of linear equations, when the determinant of the coefficient matrix is zero, the system has no solution.

In Exercises 6770, determine whether the statement is true or false. Justify your answer. The points and are collinear.

Writing Use your schools library, the Internet, or some other reference source to research a few current real-life uses of cryptography. Write a short summary of these uses. Include a description of how messages are encoded and decoded in each case.

Writing (a) State Cramers Rule for solving a system of linear equations. (b) At this point in the text, you have learned several methods for solving systems of linear equations. Briefly describe which method(s) you find easiest to use and which method(s) you find most difficult to use.

3. Reasoning Use determinants to find the area of a triangle with vertices and Confirm your answer by plotting the points in a coordinate plane and using the formula Area 1 2baseheight.

HOW DO YOU SEE IT? At this point in the text, you have learned several methods for finding an equation of a line that passes through two given points. Briefly describe the methods that can be used to find the equation of the line that passes through the two points shown. Discuss the advantages and disadvantages of each method.

In Exercises 14, determine the order of the matrix. 4 0 5

In Exercises 14, determine the order of the matrix. 3 2 1 7 0 1 6 4

In Exercises 14, determine the order of the matrix. 3

In Exercises 14, determine the order of the matrix. 6 2 580

In Exercises 5 and 6, write the augmented matrix for the system of linear equations. 3x 10y 15 5x  4y 22

In Exercises 5 and 6, write the augmented matrix for the system of linear equations. 8x 7y 4z 12 3x 5y 2z 20

In Exercises 7 and 8, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.) 5 4 9 1 2 4 7 0 2 9 10 3

In Exercises 7 and 8, write the system of linear equations represented by the augmented matrix. (Use variables and if applicable.) 13 4 16 10 7 4 3 3 2 1 5

In Exercises 9 and 10, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.) 0 1 2 1 2 2 1 3 2

In Exercises 9 and 10, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.) 4 3 2 8 1 10 16 2 12 0

In Exercises 1114, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve the system. (Use variables and if applicable.) 1 0 0 2 1 0 3 2 1 9 2 0 x,

In Exercises 1114, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve the system. (Use variables and if applicable.) 1 0 0 3 1 0 9 1 1 4 10 2

In Exercises 1114, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve the system. (Use variables and if applicable.) 1 0 0 5 1 0 4 2 1 1 3 4

In Exercises 1114, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve the system. (Use variables and if applicable.) 1 0 0 8 1 0 0 1 1 2 7 1 1

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 5x 4y  x  y  2 22

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 2x 5y 2 3x 7y 1

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 0.3x 0.1y  0.13 0.2x 0.3y  0.25

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 0.2x 0.1y  0.4x 0.5y  0.07 0.01

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 2x  2y 4y   3 6

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 2x  2y 4y   3 6

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 2x x   2y y 3y   z 2z 2z    7 4 3

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 2x x   2y y 3y   z 2z 2z    4 24 20

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 2x  y 2x 2y 2x y 2z 4 5 6z 2

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 2y  6z  2x 5y 15z  3x  y  3z  1 4 6

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 2x 3y  z  2x 3y 3z  4x 2y 3z  10 22 2

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 2x 3y  3z  3 6x 6y 12z 13 12x 9y z  2

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. 2x 3x x   y 2y 3y    z 3z 2z z  w  2w  3w  6 9 11 14

In Exercises 1528, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. x 4x 2x   2y  3y  4y   3z z z w 3 0 2w 0 3

In Exercises 2934, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. x x 2x   2y y y  z z 3z    3 3 10

In Exercises 2934, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. x 3x x  3y y y  z z 3z    2 6 2 x

In Exercises 2934, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. x  y 2z  2x 3y  z  5x 4y 2z  1 2 4

In Exercises 2934, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. 4x 4y 4z 5 4x 2y 8z 1 5x 3y 8z 6

In Exercises 2934, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. 2x y 9z  x 3y 4z  5x 2y z  8 15 17 4

In Exercises 2934, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. 3x  y 7z  5x 2y z  x  y 4z  20 34 8 2

In Exercises 35 and 36, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. 3x y 5z 2w  x 6y 4z w  5x y  z 3w  4y z 8w  44 1 15 58 978113

In Exercises 35 and 36, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. 4x 12y 2z  x  6y 4z  x  6y  z  2x 10y 2z  20 12 8 10

In Exercises 3740, find x and y 1 y x 9 1 7 12 9 x

In Exercises 3740, find x and y 1 x 4 0 5 y  1 8 4 0 5 0

In Exercises 3740, find x and y x 3 0 2 4 3 y 5 4y 2 6x  5x 1 0 2 4 3 16 44 2 6 1

In Exercises 3740, find x and y 9 0 6 4 3 1 2 7 1 5 4 0 x 9 0 1 2x 4 3 1 x 10 7 1 5 2y 0 

In Exercises 4144, if possible, find (a) (b) (c) and (d) A B, A B, 4A, A 3B. A 2 3 2 5, A B  3 12 10 8 A

In Exercises 4144, if possible, find (a) (b) (c) and (d) A B, A B, 4A, A 3B. A 4 6 10 3 1 1 B  3 15 20 11 25 29A

In Exercises 4144, if possible, find (a) (b) (c) and (d) A B, A B, 4A, A 3B. A 5 7 11 4 2 2 B  0 4 20 3 12 40A

In Exercises 4144, if possible, find (a) (b) (c) and (d) A B, A B, 4A, A 3B. A 6 5 7, B  1 4 8

In Exercises 4550, evaluate the expression. If it is not possible, explain why. 7 1 3 5 10 14 20 3 B 

In Exercises 4550, evaluate the expression. If it is not possible, explain why. 11 7 16 2 19 16 8 2 0 4 10 7 1

In Exercises 4550, evaluate the expression. If it is not possible, explain why. 2 1 5 6 2 4 0 8 7 1 1 1 2 4

In Exercises 4550, evaluate the expression. If it is not possible, explain why. 8 2 0 1 4 6 8 12 05 2 3 6 0 1 12 4 1 8 2 1 5

In Exercises 4550, evaluate the expression. If it is not possible, explain why. 3 8 1 2 3 5 16 4 2 2 7 3 6 8 2

In Exercises 4550, evaluate the expression. If it is not possible, explain why. 5 2 7 8 0 2 2 4 4 6 1 2 11 3 3 8

In Exercises 5154, solve for in the equation, where X 2A 3B

In Exercises 5154, solve for in the equation, where 6X 4A 3B

In Exercises 5154, solve for in the equation, where 3X 2A B

In Exercises 5154, solve for in the equation, where 2A 5B 3X

In Exercises 5558, find if possible. A 2 3 2 5, A B  3 12 10 8 A 2 3

In Exercises 5558, find if possible. A 5 7 11 4 2 2 B  4 20 15 12 40 30A

In Exercises 5558, find if possible. A 5 7 11 4 2 2 , B  4 20 12 40

In Exercises 5558, find if possible. A 6 5 7, B  1 4 8

In Exercises 5962, evaluate the expression. If it is not possible, explain why. 1 5 6 2 4 0 6 4 2 0 8 0 B

In Exercises 5962, evaluate the expression. If it is not possible, explain why. 1 2 5 4 6 06 4 2 0 8 0 1

In Exercises 5962, evaluate the expression. If it is not possible, explain why. 1 2 5 4 6 06 2 8 4 0 0

In Exercises 5962, evaluate the expression. If it is not possible, explain why. 1 0 0 3 2 0 2 4 3 4 0 0 3 3 0 2 1 2

In Exercises 6366, use the matrix capabilities of a graphing utility to find the product, if possible. 4 11 12 1 7 3 3 2 5 2 6 2 9781

In Exercises 6366, use the matrix capabilities of a graphing utility to find the product, if possible. 2 4 3 2 10 2 1 5 3 1 2 2

In Exercises 6366, use the matrix capabilities of a graphing utility to find the product, if possible. 1 0 1 2 4 1 1 2 3 1 1 2

In Exercises 6366, use the matrix capabilities of a graphing utility to find the product, if possible. 4 2 6 2 0 2 1 3 0

Manufacturing A tire corporation has three factories, each of which manufactures two models of tires. The production levels are represented by Factory 12 3 Model Find the production levels when production is decreased by 5%.

Manufacturing A power tool company has four manufacturing plants, each of which produces three types of cordless power tools. The production levels are represented by Plant 1 23 4 Type Find the production levels when production is increased by 20%.

Manufacturing An electronics manufacturing company produces three different models of headphones that are shipped to two warehouses. The shipment levels are represented by Warehouse 1 2 Model The prices per unit are represented by the matrix Compute and interpret the result.

Cell Phone Charges The pay-as-you-go charges (in dollars per minute) of two cellular telephone companies for calls inside the coverage area, regional roaming calls, and calls outside the coverage area are represented by Company A B Coverage area The numbers of minutes you plan to use in the coverage areas per month are represented by the matrix Compute and interpret the result.

In Exercises 7174, show that is the inverse of A A 4 7 1 2, B B  2 7 1 4 A 4 7

In Exercises 7174, show that is the inverse of A A 5 11 1 2, B B  2 11 1 5 A

In Exercises 7174, show that is the inverse of A A  1 1 6 1 0 2 0 1 3 B  2 3 2 3 3 4 1 1 1 A 

In Exercises 7174, show that is the inverse of A A 1 1 8 1 0 4 0 1 2 , B B  2 3 2 1 1 2 1 2 1 2 1 2 A 

In Exercises 7578, find the inverse of the matrix (if it exists). 6 5 5 4 B

In Exercises 7578, find the inverse of the matrix (if it exists). 3 2 5 3

In Exercises 7578, find the inverse of the matrix (if it exists). 2 1 2 0 1 2 3 1 1

In Exercises 7578, find the inverse of the matrix (if it exists). 0 5 7 2 2 3 1 3 4 2

In Exercises 7982, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 1 3 1 2 7 4 2 9 7

In Exercises 7982, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 1 2 1 4 3 18 6 1 16 1

In Exercises 7982, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 1 4 3 1 3 4 4 2 1 2 1 1 6 6 2 2 1

In Exercises 7982, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 8 4 1 1 0 2 2 4 2 0 1 1 8 2 4 1 1

In Exercises 8390, use the formula on page 527 to find the inverse of the matrix (if it exists). 7 8 2 2 2

In Exercises 8390, use the formula on page 527 to find the inverse of the matrix (if it exists). 10 7 4 3

In Exercises 8390, use the formula on page 527 to find the inverse of the matrix (if it exists). 12 10 6 5

In Exercises 8390, use the formula on page 527 to find the inverse of the matrix (if it exists). 18 6 15 5 12

In Exercises 8390, use the formula on page 527 to find the inverse of the matrix (if it exists). 1 2 3 10 20 6

In Exercises 8390, use the formula on page 527 to find the inverse of the matrix (if it exists). 3 4 4 5 5 2 8 3 1 2

In Exercises 8390, use the formula on page 527 to find the inverse of the matrix (if it exists). 0.5 0.2 0.1 0.4

In Exercises 8390, use the formula on page 527 to find the inverse of the matrix (if it exists). 1.6 1.2 3.2 2.4 0.

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. x 4y  2x 7y  8 5

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 5x y  9x 2y  13 24

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 3x 10y  5x 17y  8 13

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 4x 2y  19x 9y  10 47

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 1 2x 3x   1 3y 2y   2 0

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 5 6x 4x  3 8y 3y   2 0

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 25 0.3x 0.4x   0.7y 0.6y   10.2 7.6

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 3.5x 2.5x 4.5y 7.5y   8 25

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 3x 2y z  x y 2z  5x  y  z  6 1 7

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 4x  3x  2x  5y 2y  y  6z  2z  z  6 8 3

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 2x  y 2z  x 4y  z  y z  13 11 0 4

In Exercises 91102, use an inverse matrix to solve (if possible) the system of linear equations. 3x y 5z  x  y 6z  8x 4y z  14 8 44

In Exercises 103108, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. x 2y  1 3x 4y  5

In Exercises 103108, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. x 3y  6x 2y  23 18

In Exercises 103108, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. 6 5x 12 5 x  4 7y 12 7 y   6 5 17 5

In Exercises 103108, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. 5x 2x   10y y   7 98

In Exercises 103108, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. 3x 3y 4z  y  z  4x 3y 4z  2 1 1

In Exercises 103108, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. x 3y 2z  2x 7y 3z  x y 3z  8 19 3 3

In Exercises 109112, find the determinant of the matrix. 8 2 5 4

In Exercises 109112, find the determinant of the matrix. 9 7 11 4 8

In Exercises 109112, find the determinant of the matrix. 50 10 30 5

In Exercises 109112, find the determinant of the matrix. 14 12 24 15 50

In Exercises 113116, find all the (a) minors and (b) cofactors of the matrix. 2 7 1 4

In Exercises 113116, find all the (a) minors and (b) cofactors of the matrix. 3 5 6 4 2 7

In Exercises 113116, find all the (a) minors and (b) cofactors of the matrix. 3 2 1 2 5 8 1 0 6

In Exercises 113116, find all the (a) minors and (b) cofactors of the matrix. 8 6 4 3 5 1 4 9 2 3

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 2 2 1 0 1 1 0 0 3

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 0 0 1 1 1 1 2 2 3 2

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 4 2 1 1 3 1 1 2 0

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 1 2 5 2 3 1 1 0 3 4 2

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 2 6 5 4 0 3 1 2 4 1

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 1 4 0 1 1 1 4 2 1 2

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 1 1 2 2 2 2 4 0 1 4 3 0 0 1 1 0 1

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 1 4 2 0 2 1 3 2 1 4 3 4 2 1 0 2 1

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 3 0 6 0 0 8 1 3 4 1 8 4 0 2 2 1

In Exercises 117126, find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. 5 0 3 1 6 1 4 6 0 1 5 0 0 2 1 3 3 0 6

In Exercises 127130, use Cramers Rule to solve (if possible) the system of equations. 5x 2y  11x 3y  6 23 7

In Exercises 127130, use Cramers Rule to solve (if possible) the system of equations. 3x 8y  9x 5y  7 37

In Exercises 127130, use Cramers Rule to solve (if possible) the system of equations. 2x 3y 5z  4x y  z  x 4y 6z  11 3 15 3

In Exercises 127130, use Cramers Rule to solve (if possible) the system of equations. 5x 2y  z  3x 3y z  2x y 7z  15 7 3 2x

In Exercises 131134, use a determinant to find the area of the triangle with the given vertices.

In Exercises 131134, use a determinant to find the area of the triangle with the given vertices.

In Exercises 131134, use a determinant to find the area of the triangle with the given vertices.

In Exercises 131134, use a determinant to find the area of the triangle with the given vertices.

In Exercises 135 and 136, use a determinant to determine whether the points are collinear. 1, 7, 3, 9, 3, 15 (4

In Exercises 135 and 136, use a determinant to determine whether the points are collinear. 0, 5, 2, 6, 8, 1 1,

In Exercises 137140, use a determinant to find an equation of the line passing through the points. 4, 0, 4, 4

In Exercises 137140, use a determinant to find an equation of the line passing through the points. 2, 5, 6, 1

In Exercises 137140, use a determinant to find an equation of the line passing through the points. 0.8, 0.2, 0.7, 3.25 2, 3, 7 2, 1

In Exercises 137140, use a determinant to find an equation of the line passing through the points. 0.8, 0.2, 0.7, 3.2

In Exercises 141 and 142, (a) write the uncoded row matrices for the message and then (b) encode the message using the encoding matrix. LOOK OUT BELOW 2 3 6 2 0 2 0 3 3

In Exercises 141 and 142, (a) write the uncoded row matrices for the message and then (b) encode the message using the encoding matrix. HEAD DUE WEST 1 3 1 2 7 4 2 9 7

In Exercises 143 and 144, decode the cryptogram by using the inverse of the matrix 5 11 2 370 265 225 57 48 33 32 15 20 245 171 147

In Exercises 143 and 144, decode the cryptogram by using the inverse of the matrix 145 105 92 264 188 160 23 16 15 129 84 78 9 8 5 159 118 100 219 152 133 370 265 225 105 84 63

In Exercises 145 and 146, determine whether the statement is true or false. Justify your answer It is possible to find the determinant of a matrix.

In Exercises 145 and 146, determine whether the statement is true or false. Justify your answer a11 a21 a31 c1 a12 a22 a32 c2 a13 a23 a33 c3 a11 a21 a31 a12 a22 a32 a13 a23 a33  a11 a21 c1 a12 a22 c2 a13 a23 c3

Using a Graphing Utility Use the matrix capabilities of a graphing utility to find the inverse of the matrix What message appears on the screen? Why does the graphing utility display this message?

Invertible Matrices Under what conditions does a matrix have an inverse?

Writing What is meant by the cofactor of an entry of a matrix? How are cofactors used to find the determinant of the matrix?

Think About It Three people were asked to solve a system of equations using an augmented matrix. Each person reduced the matrix to row-echelon form. The reduced matrices were and Can all three be right? Explain

Think About It Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has a unique solution.

Solving an Equation Solve the equation for  2  3 5 8 0 .