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
ANSWERED
asked 2021-02-19

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 \(AB = BA\)

Answers (1)

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 \(AB \neq BA\)

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