5.1. MATRIX ARITHMETIC 89

and so, from the way we multiply a matrix times a vector,

(Abk)i = ∑r

Air (bk)r = ∑r

AirBrk

Thus, the ith entry from the top of Abk is the ith entry in the kth column of AB showing thatindeed the claim is true. ■

Example 5.1.16 Multiply if possible

 1 23 12 6

( 2 3 17 6 2

).

First check to see if this is possible. It is of the form (3×2)(2×3) and since the insidenumbers match, the two matrices are conformable and it is possible to do the multiplication.The result should be a 3×3 matrix. The answer is of the form

 1 23 12 6

( 27

),

 1 23 12 6

( 36

),

 1 23 12 6

( 12

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

13 15 546 42 14

 ,

a 3×3 matrix as desired. In terms of the i jth 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 i jth

entries works for other entries.

Example 5.1.17 Multiply if possible

 1 23 12 6

 2 3 1

7 6 20 0 0

 .

This is not possible because it is of the form (3×2)(3×3) and the middle numbersdon’t match. In other words the two matrices are not conformable in the indicated order.

Example 5.1.18 Multiply if possible

 2 3 17 6 20 0 0

 1 2

3 12 6

 .

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

29 320 0

 .

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