Linear Algebra with Applications, 10th Edition

1.4 Matrix Algebra

The algebraic rules used for real numbers may or may not work when matrices are used. For example, if a and b are real numbers, then

$$\begin{array}{ccc}a+b=b+a& \text{and}& ab=ba\end{array}$$

For real numbers, the operations of addition and multiplication are both commutative. The first of these algebraic rules works when we replace a and b by square matrices A and B, that is,

$$A+B=B+A$$

However, we have already seen that matrix multiplication is not commutative. This fact deserves special emphasis.

WARNING: In general, $AB\ne BA$. Matrix multiplication is not commutative.

In this section, we examine which algebraic rules work for matrices and which do not.

Algebraic Rules

The following theorem provides some useful rules for doing matrix algebra.

Theorem 1.4.1

Each ...