Section 7.4

Matrix Algebra

If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding entries of A and B. The difference of A and B, denoted A - B, is obtained by subtracting corresponding entries of A and B.

1 2 2 3 0 4 and

0 1 3 2 1 4A B

102622

BA

The Zero Matrix

If A is an m × n matrix and s is a scalar, then we let kA denote the matrix obtained by multiplying every element of A by k. This procedure is called scalar multiplication.

1 2 2 0 2 3 3 10 1 3 1 2 1 0 4

A B C

2

Find if 1 2 4 and 13

RC R C

Find the product if2 4

3 2 11 3

0 4 13 1

AB

A B

Find the product if2 4

3 2 11 3

0 4 13 1

BA

A B

2 43 2 1

1 30 4 1

3 1B A

Recall from last example: 5 71 11

AB

410941432126

BA

1 3 2 0 and

2 7 3 4A B

7 12 2 6

17 28 5 19AB BA

AB ≠ BA

Section 7.6 Solutions of Linear Systems by

Matrix Inverses

2 43 2 1

and 1 30 4 1

3 1A B

1

113 1 2Show that the inverse of is 4 2 32

2

A A

1 1 2The matrix 0 1 3 is nonsingular. Find its inverse.

2 2 1A

2 1Show that the matrix has no inverse.

4 2A

1 1 2 1 0 1 3 2

2 2 1 1

xA X y B AX B

z

2 1Solve the system of equations: 3 2

2 2 1

x y zy z

x y z

1

1

7 5 1 169 9 9 912 1 1 523 3 3 3

12 4 1 119 9 9 9

X A B

A B

A)

B)

C)

D)

A)

B)

C)

D)