13.4 Product of Two Matrices

•The product of two matrices A and B is defined provided the number of columns in A is equal to the number of rows in B.

• If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix.

State whether AB is defined. If so, give the dimensions. 1. A: 2 x 4

B: 4 x 3

2. A: 1 x 4 B: 1 x 4

3. A: 6 x 3 B: 3 x 1

•To multiply two matrices, we multiply the elements in a row of the first matrix by the elements in a column of the second matrix and add the products together. For example:

3 25 4

⎡

⎣⎢

⎤

⎦⎥⋅ −2 3

1 6⎡

⎣⎢

⎤

⎦⎥

Step 1:

•To get the element in the first row and first column of the product, we multiply the elements of the first row in the first matrix by the elements in the first column of the second matrix:

3 2⎡

⎣⎢

⎤

⎦⎥

−21

⎡

⎣⎢

⎤

⎦⎥=

3 −2( ) + 2 1( )⎡

⎣⎢⎢

⎤

⎦⎥⎥= −4⎡

⎣⎢

⎤

⎦⎥

Step 2:

•To get the element in the first row and second column of the product, we multiply the elements of the first row in the first matrix by the elements in the second column of the second matrix:

3 2⎡

⎣⎢

⎤

⎦⎥

36

⎡

⎣⎢

⎤

⎦⎥=

−4 3 3( ) + 2 6( )⎡

⎣⎢⎢

⎤

⎦⎥⎥= −4 21⎡

⎣⎢

⎤

⎦⎥

Step 3:

•To get the element in the second row and first column of the product, we multiply the elements of the second row in the first matrix by the elements in the first column of the second matrix:

5 4

⎡

⎣⎢

⎤

⎦⎥

−21

⎡

⎣⎢

⎤

⎦⎥=

−4 215 −2( ) + 4 1( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥= −4 21

−6⎡

⎣⎢

⎤

⎦⎥

Step 4:

•To get the element in the second row and second column of the product, we multiply the elements of the second row in the first matrix by the elements in the second column of the second matrix:

5 4

⎡

⎣⎢

⎤

⎦⎥

36

⎡

⎣⎢

⎤

⎦⎥=

−4 21−6 5 3( ) + 4 6( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥= −4 21

−6 39⎡

⎣⎢

⎤

⎦⎥

Find the indicated product if it is defined. If not, write “not defined”.

4. −2 8 10 −7 3

⎡

⎣⎢

⎤

⎦⎥

02−6

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Find the indicated product if it is defined. If not, write “not defined”.

5. 4 80 −31 −4

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1 0−2 5

⎡

⎣⎢

⎤

⎦⎥

Find the indicated product if it is defined. If not, write “not defined”.

6. 10

⎡

⎣⎢

⎤

⎦⎥

1 23 −4

⎡

⎣⎢

⎤

⎦⎥

Find the indicated product if it is defined. If not, write “not defined”.

7.

−1 33 4

⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜

⎞

⎠⎟

2