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

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

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

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 B is the inverse of A. A    2 5 1 3, B   3 5 1 2A   

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

In Exercises 512, show that B is the inverse of A. A    1 3 2 4, B  1 24 3 2 1

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

In Exercises 512, show that B 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 B is the inverse of A. A    4 1 0 1 2 1 B  1 42 1 1 4 4 4 6 11 7

In Exercises 512, show that B is the inverse of A. A   2 3 1 3 0 0 1 1 2 0 2 1 1 1 1 0 B  1 31 2 1 3 3 9 0 6 2 7 1 6 2 10 1 6

In Exercises 512, show that B 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

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

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 

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). 1 3 1 2 7 4 2 9 7 

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).

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

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 

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). 12 1 0 34 0 1 14 3212 

In Exercises 2534, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). 56 0 1 13 23 12 11 6 2 52  

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

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). 1 3 2 1 2 5 5 4 1 2 2 4 2 3 5 11

In Exercises 3540, use the formula on page 569 to find the inverse of the matrix (if it exists).   2 1 3 5

In Exercises 3540, use the formula on page 569 to find the inverse of the matrix (if it exists).  1 3 2 2 

In Exercises 3540, use the formula on page 569 to find the inverse of the matrix (if it exists).   4 2 6 3

In Exercises 3540, use the formula on page 569 to find the inverse of the matrix (if it exists). 12 5 3 2 

In Exercises 3540, use the formula on page 569 to find the inverse of the matrix (if it exists).   72 15 34 45 

In Exercises 3540, use the formula on page 569 to find the inverse of the matrix (if it exists). 14 53 9 4 89

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

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

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

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

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

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations. 14 x  38 y  2 32 x  34 y  12

In Exercises 4956, use an inverse matrix to solve (if possible) the system of linear equations. 56 x  43 x  y  20 72 y  51

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

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

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

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

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

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.  x 0.065x   y  0.07y  2y  z  0.09z  z  total investment annual return 0 Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. $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.  x 0.065x   y  0.07y  2y  z  0.09z  z  total investment annual return 0 Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. $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.  x 0.065x   y  0.07y  2y  z  0.09z  z  total investment annual return 0 Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. $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.  x 0.065x   y  0.07y  2y  z  0.09z  z  total investment annual return 0 Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond. $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.

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.

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.

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.

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 loam 750 units of loam 400 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 70. 500 units of sand 500 units of loam 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 2x2 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, AX  B, 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

Let be the matrix Use the determinant of to state the conditions for which (a) A1 exists and (b) A1  A.