2.1. MATRICES 43

answer is of the form 1 2

3 1

2 6

( 2

7

),

 1 2

3 1

2 6

( 3

6

),

 1 2

3 1

2 6

( 1

2

)where the commas separate the columns in the resulting product. Thus the above productequals  16 15 5

13 15 5

46 42 14

 ,

a 3× 3 matrix as desired. In terms of the ijth entries and the above definition, the entry inthe third row and second column of the product should equal∑

j

a3kbk2 = a31b12 + a32b22 = 2× 3 + 6× 6 = 42.

You should try a few more such examples to verify the above definition in terms of the ijth

entries works for other entries.

Example 2.1.10 Multiply if possible

 1 2

3 1

2 6

 2 3 1

7 6 2

0 0 0

 .

This is not possible because it is of the form (3× 2) (3× 3) and the middle numbersdon’t match.

Example 2.1.11 Multiply if possible

 2 3 1

7 6 2

0 0 0

 1 2

3 1

2 6

 .

This is possible because in this case it is of the form (3× 3) (3× 2) and the middlenumbers do match. When the multiplication is done it equals 13 13

29 32

0 0

 .

Check this and be sure you come up with the same answer.

Example 2.1.12 Multiply if possible

 1

2

1

( 1 2 1 0).

In this case you are trying to do (3× 1) (1× 4) . The inside numbers match so you cando it. Verify  1

2

1

( 1 2 1 0)=

 1 2 1 0

2 4 2 0

1 2 1 0

