Page 45 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The definition of matrix multiplication requires that the number of columns of the first factor A be the same as the number of
rows of the second factor B in order to form the product . If this condition is not satisfied, the product is undefined. A
convenient way to determine whether a product of two matrices is defined is to write down the size of the first factor and, to
the right of it, write down the size of the second factor. If, as in 3, the inside numbers are the same, then the product is
defined. The outside numbers then give the size of the product.

(3)

EXAMPLE 6 Determining Whether a Product Is Defined
Suppose that A, B, and C are matrices with the following sizes:

Then by 3, is defined and is a matrix; is defined and is a matrix; and is defined and is a matrix.
The products , , and are all undefined.

In general, if is an matrix and is an matrix, then, as illustrated by the shading in 4,

(4)

the entry in row i and column j of is given by

(5)

Partitioned Matrices

A matrix can be subdivided or partitioned into smaller matrices by inserting horizontal and vertical rules between selected
rows and columns. For example, the following are three possible partitions of a general matrix A—the first is a partition
of A into four submatrices , , , and ; the second is a partition of A into its row matrices , , and ; and the
third is a partition of A into its column matrices , , , and :