8. Matrices and Determinants

In Exercises 1 - 4, a. Give the order of each matrix, b. If A = [a_ij], identify a_32 and a_23, or explain why identification is not possible, 4 - 7 5 - 6 8 - 1 (please enclose the values above in a matrix symbol)

In Exercises 9 - 16, find the following matrices: b. A - B 4 1 5 9 A = B = 3 2 0 7

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1.

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 4 1 5 9 A = B = 3 2 0 7

In Exercises 9 - 16, find the following matrices: a. A + B 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4

In Exercises 9 - 16, find the following matrices: c. - 4A 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4

In Exercises 9 - 16, find the following matrices: a. A + B A = [6 2 - 3], B = [4 - 2 3]

In Exercises 9 - 16, find the following matrices: c. - 4A A = [6 2 - 3], B = [4 - 2 3]

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 2 - 10 - 2 6 10 - 2 A = 14 12 10 B = 0 - 12 - 4 4 - 2 2 - 5 2 - 2

Find the values of the variables for which each statement is true, if possible. See Examples 1 and 2. =

In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X. 2X + A = B

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 A = [1 2 3 4], B = 3 4

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 4 2 2 3 4 A = 6 1 B = 3 5 - 1 - 2 0

Find each sum or difference, if possible. See Examples 2 and 3. <1x4 Matrix> - <1x4 Matrix>

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason. 4 0 5 1 1 - 1 A = - 3 5 B = C = 0 1 - 2 - 2 - 1 1 A(BC)

Let A = and B = . Find each of the following. See Examples 2 –4. (3/2)B

Find each product, if possible. See Examples 5–7. <4x2 Matrix>

Find the values of the variables for which each statement is true, if possible. [2x2 matrix] = [2x2 matrix]

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1.

In Exercises 9 - 16, find the following matrices: a. A + B 4 1 5 9 A = B = 3 2 0 7

In Exercises 9 - 16, find the following matrices: c. - 4A 4 1 5 9 A = B = 3 2 0 7

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 1 3 2 - 1 A = 3 4 B = 3 - 2 5 6 0 1

In Exercises 9 - 16, find the following matrices: b. A - B 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 2 - 5 A = - 4 B = 3 1 - 1

In Exercises 9 - 16, find the following matrices: b. A - B A = [6 2 - 3], B = [4 - 2 3]

In Exercises 9 - 16, find the following matrices: c. - 4A 2 - 10 - 2 6 10 - 2 A = 14 12 10 B = 0 - 12 - 4 4 - 2 2 - 5 2 - 2

Find the values of the variables for which each statement is true, if possible. See Examples 1 and 2. <1x3 Matrix> = <1x2 Matrix>

In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X. X - A = B

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 3 3 - 2 A = B = 5 3 - 1 6

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 - 1 4 1 1 0 A = 4 - 1 3 B = 1 2 4 2 0 - 2 1 - 1 3

Find each sum or difference, if possible. See Examples 2 and 3. <3x2 Matrix> - <3x2 Matrix>

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason. 4 0 5 1 1 - 1 A = - 3 5 B = C = 0 1 - 2 - 2 - 1 1 A - C

Let A = and B = . Find each of the following. See Examples 2 –4. 2A

Find each product, if possible. See Examples 5–7.

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

Find AB and BA for the following matrices. A= and B = Matrix B acts as the multiplicative element for 2 * 2 square matrices.

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. x 2y 4 12 = z 9 3 9

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1.

In Exercises 9 - 16, find the following matrices: c. - 4A 1 3 2 - 1 A = 3 4 B = 3 - 2 5 6 0 1

In Exercises 9 - 16, find the following matrices: c. - 4A 2 - 5 A = - 4 B = 3 1 - 1

In Exercises 9 - 16, find the following matrices: b. A - B 2 - 10 - 2 6 10 - 2 A = 14 12 10 B = 0 - 12 - 4 4 - 2 2 - 5 2 - 2

Find the values of the variables for which each statement is true, if possible. See Examples 1 and 2. =

In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X. 4A + 3B = - 2X

Find each sum or difference, if possible. See Examples 2 and 3. <1x2 Matrix> - <1x3 Matrix>

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason. 4 0 5 1 1 - 1 A = - 3 5 B = C = 0 1 - 2 - 2 - 1 1 BC + CB

Find each product, if possible. See Examples 5–7. <3x2 Matrix><2x3 Matrix>

For each pair of matrices A and B, find (a) AB and (b) BA. See Example 7. A= , B =

Perform each operation, if possible. [3x3 matrix] [3x3 matrix]

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. x 6 = 4 y

In Exercises 9 - 16, find the following matrices: b. A - B 1 3 2 - 1 A = 3 4 B = 3 - 2 5 6 0 1

In Exercises 9 - 16, find the following matrices: b. A - B 2 - 5 A = - 4 B = 3 1 - 1

Find the values of the variables for which each statement is true, if possible. See Examples 1 and 2. =

In Exercises 9 - 16, find the following matrices: a. A + B 2 - 10 - 2 6 10 - 2 A = 14 12 10 B = 0 - 12 - 4 4 - 2 2 - 5 2 - 2

In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X. B - X = 4A

Find each sum or difference, if possible. See Examples 2 and 3. <3x1 Matrix> - <3x1 Matrix>

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason. 4 0 5 1 1 - 1 A = - 3 5 B = C = 0 1 - 2 - 2 - 1 1 4B - 3C

Find each product, if possible. See Examples 5–7. <3x2 Matrix><1x3 Matrix>

Perform each operation, if possible. [3x2 matrix] [2x3 matrix]

For each pair of matrices A and B, find (a) AB and (b) BA. See Example 7. A= , B =

Fill in the blank(s) to correctly complete each sentence. For the following statement to be true, the value of x must be ____, and the value of y must be ____. =

In Exercises 1 - 4, a. Give the order of each matrix, b. If A = [a_ij], identify a_32 and a_23, or explain why identification is not possible, 1 - 5 π e 0 7 - 6 - π - 2 1/2 11 - 1/5 (please enclose the values above in a matrix symbol)

In Exercises 9 - 16, find the following matrices: a. A + B 1 3 2 - 1 A = 3 4 B = 3 - 2 5 6 0 1

In Exercises 9 - 16, find the following matrices: c. - 4A 1 3 2 - 1 A = 3 4 B = 3 - 2 5 6 0 1

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1. <1x1 Matrix>

In Exercises 9 - 16, find the following matrices: a. A + B 2 - 5 A = - 4 B = 3 1 - 1

In Exercises 9 - 16, find the following matrices: c. - 4A 2 - 5 A = - 4 B = 3 1 - 1

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B A = [6 2 - 3], B = [4 - 2 3]

In Exercises 9 - 16, find the following matrices: b. A - B 2 - 10 - 2 6 10 - 2 A = 14 12 10 B = 0 - 12 - 4 4 - 2 2 - 5 2 - 2

In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X. 3X + 2A = B

In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X. 4A + 3B = - 2X

Find each sum or difference, if possible. See Examples 2 and 3. <3x1 Matrix> + <1x3 Matrix>

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 2 - 3 1 - 1 - 1 1 A = B = 1 1 - 2 1 5 4 10 5

Find each product, if possible. See Examples 5–7. <1x2 Matrix>

Given A = , B = <2x3 Matrix>, and C = <3x2 Matrix>, find each product, if possible. See Examples 5–7. AB

Perform each operation, if possible. [2x2 matrix] [2x2 matrix]

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. x 6 = 4 y

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1. <1x2 Matrix>

In Exercises 9 - 16, find the following matrices: b. A - B 1 3 2 - 1 A = 3 4 B = 3 - 2 5 6 0 1

In Exercises 9 - 16, find the following matrices: b. A - B 2 - 5 A = - 4 B = 3 1 - 1

In Exercises 9 - 16, find the following matrices: a. A + B 2 - 10 - 2 6 10 - 2 A = 14 12 10 B = 0 - 12 - 4 4 - 2 2 - 5 2 - 2

In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X. B - X = 4A

Find each sum or difference, if possible. See Examples 2 and 3. <2x3 Matrix> + <2x3 Matrix>

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason. 4 0 5 1 1 - 1 A = - 3 5 B = C = 0 1 - 2 - 2 - 1 1 4B - 3C

Let A = and B = . Find each of the following. See Examples 2 –4. -A + (1/2)B

Find each product, if possible. See Examples 5–7.

Given A = , B = <2x3 Matrix>, and C = <3x2 Matrix>, find each product, if possible. See Examples 5–7. BC

Perform each operation, if possible. [3x2 matrix] - [2x2 matrix]

In Exercises 9 - 16, find the following matrices: c. - 4A 4 1 5 9 A = B = 3 2 0 7

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1. <5x1 Matrix>

In Exercises 9 - 16, find the following matrices: a. A + B 1 3 2 - 1 A = 3 4 B = 3 - 2 5 6 0 1

In Exercises 9 - 16, find the following matrices: a. A + B 2 - 5 A = - 4 B = 3 1 - 1

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B A = [6 2 - 3], B = [4 - 2 3]

Find the values of the variables for which each statement is true, if possible. See Examples 1 and 2. <2x3 Matrix> + <2x3 Matrix> = <2x3 Matrix>

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 - 1 4 1 1 0 A = 4 - 1 3 B = 1 2 4 2 0 - 2 1 - 1 3

Let A = and B = . Find each of the following. See Examples 2 –4. 2A + 4B

Find each product, if possible. See Examples 5–7. <3x1 Matrix>

Given A = , B = <2x3 Matrix>, and C = <3x2 Matrix>, find each product, if possible. See Examples 5–7. BA

Perform each operation, if possible. [1x3 matrix] - [1x3 matrix] + [1x3 matrix]