5.1. MATRIX ARITHMETIC 87

First check if it is possible. This is of the form (3×3)(2×3) . The inside numbersdo not match and so you can’t do this multiplication. This means that anything you writewill be absolute nonsense because it is impossible to multiply these matrices in this order.Aren’t they the same two matrices considered in the previous example? Yes they are.It is just that here they are in a different order. This shows something you must alwaysremember about matrix multiplication.

Order Matters!

Matrix Multiplication Is Not Commutative!

This is very different than multiplication of numbers!

5.1.3 The i jth Entry Of A ProductIt is important to describe matrix multiplication in terms of entries of the matrices. What isthe i jth entry of AB? It would be the ith entry of the jth column of AB. Thus it would be theith entry of Ab j. Now

b j =

B1 j

...Bn j

and from the above definition, the ith entry is

n

∑k=1

AikBk j ≡ Ai1B1 j +Ai2B2 j + · · ·+AinBn j (5.11)

In terms of pictures of the matrix, you are doingA11 A12 · · · A1n

A21 A22 · · · A2n...

......

Am1 Am2 · · · Amn



B11 B12 · · · B1p

B21 B22 · · · B2p...

......

Bn1 Bn2 · · · Bnp

Then as explained above, the jth column is of the form

A11 A12 · · · A1n

A21 A22 · · · A2n...

......

Am1 Am2 · · · Amn



B1 j

B2 j...

Bn j

which is a m×1 matrix or column vector which equals

A11

A21...

Am1

B1 j +

A12

A22...

Am2

B2 j + · · ·+

A1n

A2n...

Amn

Bn j.