8. Matrices and Determinants

What expression in x represents ?

Evaluate each determinant in Exercises 1–10. - 4 1 5 6

Evaluate each determinant in Exercises 1–10. - 7 14 2 - 4

Evaluate each determinant. See Example 1.

Evaluate each determinant in Exercises 1–10. 1/2 1/2 1/8 - 3/4

For Exercises 11–22, use Cramer's Rule to solve each system. x + y = 7 x - y = 3

Evaluate each determinant. See Example 1.

For Exercises 11–22, use Cramer's Rule to solve each system. 4x - 5y = 17 2x + 3y = 3

For Exercises 11–22, use Cramer's Rule to solve each system. x + 2y = 3 3x - 4y = 4

Evaluate each determinant. See Example 3.

For Exercises 11–22, use Cramer's Rule to solve each system. 2x = 3y + 2 5x = 51 - 4y

In Exercises 23–30, use expansion by minors to evaluate each determinant. 3 0 0 2 1 - 5 2 5 - 1

Evaluate each determinant. See Example 3.

In Exercises 23–30, use expansion by minors to evaluate each determinant. 1 1 1 2 2 2 - 3 4 - 5

In Exercises 23–30, use expansion by minors to evaluate each determinant. 0.5 7 5 0.5 3 9 0.5 1 3

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. 1 5 6 1 4 5 1 9 10

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. 0.5 7 5 0.5 3 9 0.5 1 3

Evaluate each determinant.

In Exercises 37–44, use Cramer's Rule to solve each system. 4x - 5y - 6z = - 1 x - 2y - 5z = - 12 2x - y = 7

In Exercises 37–44, use Cramer's Rule to solve each system. x + y + z = 4 x - 2y + z = 7 x + 3y + 2z = 4

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system. 2x - 3y + 2z = 4 2x + 3y - 2z = 6 2x - 9y + 6z = 2

In Exercises 46–51, evaluate each determinant.

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system. 4x - 3y - 2z = 12 8x - 6y - 4z = 22

Use the determinant theorems to evaluate each determinant. See Example 4.

Evaluate each determinant in Exercises 49–52. - 2 - 3 3 5 1 - 4 0 0 1 2 2 - 3 2 0 1 1

In Exercises 46–51, evaluate each determinant.

Use the determinant theorems to evaluate each determinant. See Example 4.

In Exercises 53–54, evaluate each determinant. | | 3 1| |7 0| | | |- 2 3| |1 5| | | | | | 3 0| |9 - 6| | | | 0 7| |3 5| |

In Exercises 52–55, use Cramer's Rule to solve each system.

Use the determinant theorems to evaluate each determinant. See Example 4.

Use the determinant theorems to evaluate each determinant. See Example 4.

In Exercises 57–60, solve each equation for x. |- 2 x| | | = 32 | 4 6|

Use the determinant theorems to evaluate each determinant. See Example 4.

In Exercises 57–60, solve each equation for x. |1 x - 2| |3 1 1| = - 8 |0 - 2 2|

Use the determinant theorems to evaluate each determinant. See Example 4.

Use the determinant theorems to evaluate each determinant. See Example 4.

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. x + y = 4 2x - y = 2

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 4x + 3y = -7 2x + 3y = -11

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 5x + 4y = 10 3x - 7y = 6

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 1.5x + 3y = 5 2x + 4y = 3

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. x + 2y + 3z = 4 4x + 3y + 2z = 1 -x - 2y - 3z = 0

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. -2x - 2y + 3z = 4 5x + 7y - z = 2 2x + 2y - 3z = -4

What is the value of ?

Evaluate each determinant in Exercises 1–10. 5 7 2 3

Evaluate each determinant. See Example 1.

Evaluate each determinant in Exercises 1–10. - 5 - 1 - 2 - 7

Evaluate each determinant. See Example 1.

For Exercises 11–22, use Cramer's Rule to solve each system. 12x + 3y = 15 2x - 3y = 13

Find the cofactor of each element in the second row of each matrix. See Example 2.

For Exercises 11–22, use Cramer's Rule to solve each system. 3x - 4y = 4 2x + 2y = 12

Evaluate each determinant. See Example 3.

In Exercises 23–30, use expansion by minors to evaluate each determinant. 3 1 0 - 3 4 0 - 1 3 - 5

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. - 3 4 - 5 5 - 2 0 8 - 1 3

In Exercises 37–44, use Cramer's Rule to solve each system. x + y + z = 0 2x - y + z = - 1 - x + 3y - z = - 8

Evaluate each determinant.

In Exercises 37–44, use Cramer's Rule to solve each system. x + 2z = 10 2y - z = - 5 2x + 3y = 13

Solve each equation. = 2x

Evaluate each determinant in Exercises 49–52. 4 2 8 - 7 - 2 0 4 1 5 0 0 5 4 0 0 - 1

In Exercises 46–51, evaluate each determinant.

In Exercises 46–51, evaluate each determinant.

In Exercises 55–56, write the system of linear equations for which Cramer's Rule yields the given determinants. 2 - 4 8 - 4 D = D_x = 3 5 - 10 5

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. (1/2)x + (1/3)y = 2 (3/2)x - (1/2)y = -12

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 2x - y + 4z = -2 3x + 2y - z = -3 x + 4y + 2z = 17

What is the value of x if = 9?

Evaluate each determinant in Exercises 1–10. - 5 - 1 - 2 - 7

Evaluate each determinant. See Example 1.

For Exercises 11–22, use Cramer's Rule to solve each system. 12x + 3y = 15 2x - 3y = 13

Find the cofactor of each element in the second row of each matrix. See Example 2.

For Exercises 11–22, use Cramer's Rule to solve each system. 3x - 4y = 4 2x + 2y = 12

Evaluate each determinant. See Example 3.

In Exercises 23–30, use expansion by minors to evaluate each determinant. 3 1 0 - 3 4 0 - 1 3 - 5

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. - 3 4 - 5 5 - 2 0 8 - 1 3

In Exercises 37–44, use Cramer's Rule to solve each system. x + y + z = 0 2x - y + z = - 1 - x + 3y - z = - 8

Evaluate each determinant.

In Exercises 37–44, use Cramer's Rule to solve each system. x + 2z = 10 2y - z = - 5 2x + 3y = 13

Solve each equation. = 8

Evaluate each determinant in Exercises 49–52. 4 2 8 - 7 - 2 0 4 1 5 0 0 5 4 0 0 - 1

Use the determinant theorems to evaluate each determinant. See Example 4.

In Exercises 52–55, use Cramer's Rule to solve each system.

In Exercises 57–60, solve each equation for x. |1 x - 2| |3 1 1| = - 8 |0 - 2 2|

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 3x + 2y = 4 6x + 4y = 8