Page 25 - A Course in Linear Algebra with Applications

1.2: Operations with Matrices 9

which can be written more concisely using the summation
notation as

n

fc=l
Notice that the rule only makes sense if the number of
columns of A equals the number of rows of B. Also the product
of an m x n matrix and a n n x p matrix is an m x p matrix.
Example 1.2.2
Let

A = { 1 and B
1 I 2

Since A is 2 x 3 and B is 3 x 3, we see that AB is defined
and is a 2 x 3 matrix. However BA is not defined. Using the
row-times-column rule, we quickly find that

0 0 2
AB =
2 16 - 2

Example 1.2.3
Let
A = and B=
(O i) (I i

In this case both AB and BA are defined, but these matrices
are different:

AB=r Q ° ) = 0 2 2 M 1 d B A = ( ° l