8. Matrices and Determinants

In Exercises 1–8, write the augmented matrix for each system of linear equations.

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Use the given row transformation to change each matrix as indicated. See Sample 1. < 2x2 Matrix > ; -7 times row 1 added to row 2

In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

Write the augmented matrix for each system and give its dimension. Do not solve. 4x - 2y + 3z - 4 = 0 3x + 5y + z - 7 = 0 5x - y + 4z - 7 = 0

In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. -5(A+D)

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. 3x + 2y = -9 2x - 5y = -6

Solve each system, using the method indicated. 5x + 2y = -10 3x - 5y = -6 (Gauss-Jordan)

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. x + y - 5z = -18 3x - 3y + z = 6 x + 3y - 2z = -13

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Solve the system: (Hint: Let A = ln w, B = ln x, C = ln y, and D = ln z. Solve the system for A, B, C, and D. Then use the logarithmic equations to find w, x, y, and z.)

In Exercises 1–8, write the augmented matrix for each system of linear equations.

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Use the given row transformation to change each matrix as indicated. See Sample 1. < 2x2 Matrix > ; -4 times row 1 added to row 2

In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

Write the augmented matrix for each system and give its dimension. Do not solve. 2x + y + z - 3 = 0 3x - 4y + 2z + 7 = 0 x + y + z - 2 = 0

In Exercises 13–18, perform each matrix row operation and write the new matrix.

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. 3A+2D

Write the system of equations associated with each augmented matrix . Do not solve.

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. x + y = 5 x - y = -1

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. 3/8x - 1/2y = 7/8 -6x + 8y = -14

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

What is the augmented matrix of the following system? -3x + 5y = 2 6x + 2y = 7

In Exercises 1–8, write the augmented matrix for each system of linear equations.

Use the given row transformation to change each matrix as indicated. See Sample 1. < 3x3 Matrix > ; 4 times row 1 added to row 2

In Exercises 13–18, perform each matrix row operation and write the new matrix.

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. D-A

Write the system of equations associated with each augmented matrix . Do not solve. <4x3 Matrix>

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. 2x - y = 6 4x - 2y = 0

Solve for X in the matrix equation 3X+A = B where

Solve each system, using the method indicated. x - z = -3 y + z = 6 2x - 3z = -9 (Gauss-Jordan)

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Find the quadratic function f(x) = ax² + bx + c for which ƒ( − 2) = −4, ƒ(1) = 2, and f(2) = 0.

In Exercises 1–2, perform each matrix row operation and write the new matrix.

How many rows and how many columns does this matrix have? What is its dimension? <4x2 Matrix>

In Exercises 1–8, write the augmented matrix for each system of linear equations.

Use the given row transformation to change each matrix as indicated. See Sample 1. < 3x3 Matrix > ; 2 times row 1 added to row 2

In Exercises 13–18, perform each matrix row operation and write the new matrix.

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. A+D

Write the system of equations associated with each augmented matrix . Do not solve. <4x3 Matrix>

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. BD

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. 6x - 3y - 4 = 0 3x + 6y - 7= 0

Solve each system, using the method indicated. 3x + y = -7 x - y = -5 (Gaussian elimination)

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.