Question

# Determine whether each of the following statements is true or false, and explain why.If A and B are square matrices of the same size, then AB = BA

Matrices

Determine whether each of the following statements is true or false, and explain why.If A and B are square matrices of the same size, then $$AB = BA$$

2021-02-20

Given,
If A and B are square matrices of the same size, then $$AB = BA$$.
Step 2
Given statement is false.
For example,
Let $$A=\begin{bmatrix}1 & 0 \\2 & 3 \end{bmatrix} \text{ and } B=\begin{bmatrix}2 & 4 \\1 & 3 \end{bmatrix}$$
Then,
$$AB=\begin{bmatrix}1 & 0 \\2 & 3 \end{bmatrix}\begin{bmatrix}2 & 4 \\1 & 3 \end{bmatrix}$$
$$=\begin{bmatrix}2 & 4 \\7 & 17 \end{bmatrix}$$
and
$$BA=\begin{bmatrix}2 & 4 \\1 & 3 \end{bmatrix}\begin{bmatrix}1 & 0 \\2 & 3 \end{bmatrix}$$
$$=\begin{bmatrix}10 & 12 \\7 & 9 \end{bmatrix}$$
Clearly $$AB \neq BA$$