TRUE OR FALSE In Exercises 117 and 118, determinewhether the statement is true or false

  1 3 2 1 8 1 4 10 12 1 3 6

2 1 2 4 1 6 8 3 4 3 2 9 

In Exercises 3538, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.

In Exercises 3538, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.

In Exercises 3538, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.

In Exercises 3538, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.

Perform the sequence of row operations on the matrix. What did the operations accomplish? 1 2 3 2 1 1 3 4 1 (a) Add times to (b) Add times to (c) Add times to (d) Multiply by (e) Add times

Perform the sequence of row operations on the matrix. What did the operations accomplish? (a) Add to (b) Interchange and (c) Add 3 times to (d) Add times to (e) Multiply by (f ) Add the appropriate multiples of to and

In Exercises 4144, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form.   1 0 0 0 1 0 0 1 0 0 5 0

In Exercises 4144, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. 1 0 0 3 0 0 0 1 0 0 8 0 

In Exercises 4144, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form.   1 0 0 0 1 0 0 0 0 1 1 2

In Exercises 4144, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. 1 0 0 0 1 0 1 0 1 0 2 0 

In Exercises 4548, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)

In Exercises 4548, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)

In Exercises 4548, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)

In Exercises 4548, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)

In Exercises 4954, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

In Exercises 4954, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

In Exercises 4954, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

In Exercises 4954, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

In Exercises 4954, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

In Exercises 4954, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

In Exercises 5558, write the system of linear equations represented by the augmented matrix. Then use backsubstitution to solve. (Use variables and if applicable.)

In Exercises 5558, write the system of linear equations represented by the augmented matrix. Then use backsubstitution to solve. (Use variables and if applicable.)

In Exercises 5558, write the system of linear equations represented by the augmented matrix. Then use backsubstitution to solve. (Use variables and if applicable.)

In Exercises 5558, write the system of linear equations represented by the augmented matrix. Then use backsubstitution to solve. (Use variables and if applicable.)

In Exercises 5962, 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.

In Exercises 5962, 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.

In Exercises 5962, 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.

In Exercises 5962, 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.

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. x  2y  7 2x  y  8

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2x  6y  16 2x  3y  7

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 3x  2y  x  3y  27 13

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. x  y  2x  4y  4 34

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2x  6y  x  2y  22 9

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.  5x  5y  2x  3y  5 7

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 8x  4y  5x  2y  7 1

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.  x  3y  2x  6y  5 10

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. x 3x 2x  y  2y  3z   2z   z  2 5 4

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.2x  y 2y 7x  5y  3z  24  z  14  6

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. x 2x 3x    y y 2y    z z z    14 21 19

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2x x x    2y 3y y   z z    2 28 14

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.  x x   2y 4y y    3z 2z z    28 0 5

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 3x x x    2y y y    z 2z 4z    15 10 14

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.  x x   2y y   0 0

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. x 2x   2y 4y   0 0

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.  x 3x   2y 7y   z 6z   8 26

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. x 2x   y y   4z z   5 9

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. x  y  3x  4y  4x  8y  22 4 32

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.  x  2y  0 x  y  6 3x  2y  8

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 3x x 2x x     2y y y y     z 4z 2z z     w 2w w w     0 25 2 6

In Exercises 6384, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. x 3x 4x 2x     4y 2y 3y y     3z z 2z 4z     2w 4w w 3w     9 13 4 10

In Exercises 8590, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and 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 reduce the augmented matrix corresponding to the system of equations, and 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 reduce the augmented matrix corresponding to the system of equations, and 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 reduce the augmented matrix corresponding to the system of equations, and 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 reduce the augmented matrix corresponding to the system of equations, and 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 reduce the augmented matrix corresponding to the system of equations, and 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.

In Exercises 9194, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.

In Exercises 9194, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.

In Exercises 9194, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.

In Exercises 9598, use a system of equations to find the quadratic function that satisfies the equations. Solve the system using matrices.1  1, f 2  1, f 3  5

In Exercises 9598, use a system of equations to find the quadratic function that satisfies the equations. Solve the system using matrices. 1  2, f 2  9, f 3  20

In Exercises 9598, use a system of equations to find the quadratic function that satisfies the equations. Solve the system using matrices. f 2  15, f 1  7, f 1  3

In Exercises 9598, use a system of equations to find the quadratic function that satisfies the equations. Solve the system using matrices. f 2  3, f 1  3, f 2  11

In Exercises 99102, use a system of equations to find the cubic function that satisfies the equations. Solve the system using matrices.

In Exercises 99102, use a system of equations to find the cubic function that satisfies the equations. Solve the system using matrices.

In Exercises 99102, use a system of equations to find the cubic function that satisfies the equations. Solve the system using matrices.

In Exercises 99102, use a system of equations to find the cubic function that satisfies the equations. Solve the system using matrices.

Use the system to write two different matrices in row-echelon form that yield the same solution.

ELECTRICAL NETWORK The currents in an electrical network are given by the solution of the system where and are measured in amperes. Solve the system of equations using matrices.

PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices.

PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices.

FINANCE A small shoe corporation borrowed $1,500,000 to expand its line of shoes. Some of the money was borrowed at 7%, some at 8%, and some at 10%. Use a system of equations to determine how much was borrowed at each rate if the annual interest was $130,500 and the amount borrowed at 10% was 4 times the amount borrowed at 7%. Solve the system using matrices.

FINANCE A small software corporation borrowed $500,000 to expand its software line. Some of the money was borrowed at 9%, some at 10%, and some at 12%. Use a system of equations to determine how much was borrowed at each rate if the annual interest was $52,000 and the amount borrowed at 10% was times the amount borrowed at 9%. Solve the system using matrices.

TIPS A food server examines the amount of money earned in tips after working an 8-hour shift. The server has a total of $95 in denominations of $1, $5, $10, and $20 bills. The total number of paper bills is 26. The number of $5 bills is 4 times the number of $10 bills, and the number of $1 bills is 1 less than twice the number of $5 bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination.

BANKING A bank teller is counting the total amount of money in a cash drawer at the end of a shift. There is a total of $2600 in denominations of $1, $5, $10, and $20 bills The total number of paper bills is 235. The number of $20 bills is twice the number of $1 bills, and the number of $5 bills is 10 more than the number of $1 bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination

In Exercises 111 and 112, use a system of equations to find the equation of the parabola that passes through the points. Solve the system using matrices. Use a graphing utility to verify your results.

In Exercises 111 and 112, use a system of equations to find the equation of the parabola that passes through the points. Solve the system using matrices. Use a graphing utility to verify your results.

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. (x and y 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).

DATA ANALYSIS: SNOWBOARDERS The table shows the numbers of people (in millions) in the United States who participated in snowboarding in selected years from 2003 to 2007. (Source: National Sporting Goods Association) (a) Use a system of equations to find the equation of the parabola that passes through the points. Let represent the year, with corresponding to 2003. Solve the system using matrices. (b) Use a graphing utility to graph the parabola. (c) Use the equation in part (a) to estimate the number of people who participated in snowboarding in 2009. Does your answer seem reasonable? Explain. (d) Do you believe that the equation can be used for years far beyond 2007? Explain.

NETWORK ANALYSIS In Exercises 115 and 116, answer the questions about the specified network. (In a network it is assumed that the total flow into each junction is equal to the total flow out of each junction.) Water flowing through a network of pipes (in thousands of cubic meters per hour) is shown in the figure. (a) Solve this system using matrices for the water flow represented by (b) Find the network flow pattern when and (c) Find the network flow pattern when

NETWORK ANALYSIS In Exercises 115 and 116, answer the questions about the specified network. (In a network it is assumed that the total flow into each junction is equal to the total flow out of each junction.) The flow of traffic (in vehicles per hour) through a network of streets is shown in the figure. (a) Solve this system using matrices for the traffic flow represented by (b) Find the traffic flow when and (c) Find the traffic flow when

TRUE OR FALSE? In Exercises 117 and 118, determine whether the statement is true or false. Justify your answer.

TRUE OR FALSE? In Exercises 117 and 118, 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 The augmented matrix below represents system of linear equations (in variables and ) that has been reduced using Gauss-Jordan elimination. Write a system of equations with nonzero coefficients that is represented by the reduced matrix. (There are many correct answers.)

THINK ABOUT IT (a) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that is inconsistent. (b) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has an infinite number of solutions.

Describe the three elementary row operations that can be performed on an augmented matrix.

CAPSTONE In your own words, describe the difference between a matrix in row-echelon form and a matrix in reduced row-echelon form. Include an example of each to support your explanation.

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?