Question
If $\text{A}=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$ and $\text{B}=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix},$ show that AB = A and BA = B.
✓
Answer
Given, $\text{A}=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix},\text{B}=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}$
$\text{AB}=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}$
$=\begin{bmatrix}4+3-5&-4-9+10&-8-12+15\\-2-4+5&2+12-10&4+16-15\\2+3-4&-2-9+18&-4-12+12\end{bmatrix}$
$=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$
$\text{AB}=\text{A}$
$\text{BA}=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$
$=\begin{bmatrix}4+2-4&-6-8+12&-10-10+16\\-2-3+4&3+12-12&5+15-16\\2+2-3&-3-8+9&-5-10+12\end{bmatrix}$
$=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}$
$\text{BA}=\text{B}$
$\text{AB}=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}$
$=\begin{bmatrix}4+3-5&-4-9+10&-8-12+15\\-2-4+5&2+12-10&4+16-15\\2+3-4&-2-9+18&-4-12+12\end{bmatrix}$
$=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$
$\text{AB}=\text{A}$
$\text{BA}=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$
$=\begin{bmatrix}4+2-4&-6-8+12&-10-10+16\\-2-3+4&3+12-12&5+15-16\\2+2-3&-3-8+9&-5-10+12\end{bmatrix}$
$=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}$
$\text{BA}=\text{B}$
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