7. Systems of Equations & Matrices

Determinants and Cramer's Rule

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

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Determinants of 2×2 Matrices

Cramer's Rule - 2 Equations with 2 Unknowns

Determinants of 3×3 Matrices

Cramer's Rule - 3 Equations w/ 3 Unknowns

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Ex: Find the Value of a 4x4 Determinant Using Cofactor Expansion (with Zeros)

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Ex: Find the Value of a 4x4 Determinant Using Cofactor Expansion (with Zeros)

Evaluate the Determinant of a 4x4 Matrix

Determinant of a 4X4 Matrix

How To Find The Determinant of a 4x4 Matrix

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Practice this topic

Multiple Choice

Evaluate the determinant of the matrix.

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Multiple Choice

Write each equation in standard form and use Cramer's Rule to solve the system.

$y=-3x+4$

$-2x=7y-9$

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Multiple Choice

Evaluate the determinant of the matrix.

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Multiple Choice

Solve the system of equations using Cramer's Rule.

$4x+2y+3z=6$

$x+y+z=3$

$5x+y+2z=5$

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Multiple Choice

Write each equation in standard form and use Cramer's Rule to solve the system.

$y-9x=-3$

$-3x=4y-1$

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Textbook Question

What is the value of ?

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Textbook Question

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

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Textbook Question

What expression in x represents ?

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Textbook Question

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

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Textbook Question

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

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Textbook Question

What is the value of x if = 9?

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Textbook Question

Evaluate each determinant. See Example 1.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Evaluate each determinant. See Example 1.

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Textbook Question

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

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Textbook Question

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

Evaluate each determinant. See Example 1.

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Textbook Question

Evaluate each determinant. See Example 1.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Evaluate each determinant. See Example 1.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Evaluate each determinant. See Example 3.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Evaluate each determinant. See Example 3.

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Textbook Question

Evaluate each determinant. See Example 3.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Evaluate each determinant. See Example 3.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Evaluate each determinant.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Evaluate each determinant.

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Textbook Question

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

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Textbook Question

Evaluate each determinant.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

In Exercises 46–51, evaluate each determinant.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

Solve each equation. = 8

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Textbook Question

Solve each equation. = 2x

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Textbook Question

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

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Textbook Question

In Exercises 46–51, evaluate each determinant.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

In Exercises 46–51, evaluate each determinant.

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Textbook Question

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

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Textbook Question

In Exercises 46–51, evaluate each determinant.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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|

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 0 0 - 2 1 1 2 0 3 - 1 0 1 1 0 1 1 1 A = B = 0 1 - 1 0 0 1 0 1 1 0 0 - 1 1 2 0 2

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Textbook Question

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 1 2 3 7/2 - 3 1/2 A = 1 3 4 B = - 1/2 0 1/2 1 4 3 - 1/2 1 - 1/2

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Textbook Question

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 0 1 0 0 0 1 A = 0 0 1 B = 1 0 0 1 0 0 0 1 0

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Textbook Question

In Exercises 37 - 42, a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix. w - x + 2y = - 3 x - y + z = 4 - w + x - y + 2z = 2 - x + y - 2z = - 4 The inverse of is

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Textbook Question

In Exercises 37 - 42, a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix. x - y + z = 8 2y - z = - 7 2x + 3y = 1 The inverse of is

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Textbook Question

In Exercises 37 - 42, a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix. 2x + 6y + 6z = 8 2x + 7y + 6z = 10 2x + 7y + 7z = 9 The inverse of is

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Textbook Question

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. - 2 1 1 2 A = B = 3/2 - 1/2 3 4

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Textbook Question

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. - 4 0 - 2 4 A = B = 1 3 0 1

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Textbook Question

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 4 - 3 4 3 A = B = - 5 4 5 4

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Textbook Question

In Exercises 33 - 36, write each matrix equation as a system of linear equations without matrices.

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Textbook Question

In Exercises 33 - 36, write each matrix equation as a system of linear equations without matrices.

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Textbook Question

In Exercises 29 - 32, write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix. x + 3y + 4z = - 3 x + 2y + 3z = - 2 x + 4y + 3z = - 6

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Textbook Question

In Exercises 29 - 32, write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix. 6x + 5y = 13 5x + 4y = 10

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Textbook Question

In Exercises 13 - 18, use the fact that if a b d - b A = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c a each matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 10 - 2 A = - 5 1

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Textbook Question

In Exercises 13 - 18, use the fact that if a b d - b A = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c a each matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 3 - 1 A = - 4 2

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Textbook Question

In Exercises 13 - 18, use the fact that if a b d - b A = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c a each matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 2 3 A = - 1 2

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Textbook Question

In Exercises 43–44, (a) Write each linear system as a matrix equation in the form AX = B (b) Solve the system using the inverse that is given for the coefficient matrix.

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Textbook Question

In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

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Textbook Question

In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.

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Textbook Question

Answer each question. What is the product of [2x2 matrix] and I2 (in either order)?

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Answer each question. What is the product [3x3 matrix] [3x3 matrix]?

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Textbook Question

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I￬n.) [2x2 matrix] and [2x2 matrix]

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Textbook Question

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I￬n.) [3x3 matrix] and [3x3 matrix]

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Textbook Question

Find the inverse, if it exists, for each matrix. [2x2 matrix]

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Textbook Question

Find the inverse, if it exists, for each matrix. [2x2 matrix]

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Textbook Question

Find the inverse, if it exists, for each matrix. [2x2 matrix]

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Textbook Question

Find the inverse, if it exists, for each matrix. [3x3 matrix]

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Textbook Question

Find the inverse, if it exists, for each matrix. [3x3 matrix]

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Textbook Question

Find the inverse, if it exists, for each matrix. [3x3 matrix]

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