Page 43 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 3 Addition and Subtraction
Consider the matrices

Then

The expressions , , , and are undefined.

DEFINITION

If A is any matrix and c is any scalar, then the product is the matrix obtained by multiplying each entry of the matrix A
by c. The matrix is said to be a scalar multiple of A.

In matrix notation, if , then

EXAMPLE 4 Scalar Multiples
For the matrices

we have

It is common practice to denote by .

If , , …, are matrices of the same size and , , …, are scalars, then an expression of the form

is called a linear combination of , , …, with coefficients , , …, . For example, if A, B, and C are the matrices
in Example 4, then