If A= 2&-3&-5 -1&4&5 1&-3&-4 , show that A^2 = A. - Vidyadip

Question

If $\text{A}=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix},$ show that $A^2 = A$.

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Answer

Here,
$\text{A}^2=\text{AA}$
$\Rightarrow\text{A}^2=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}4+3-5&-6-12+15&-10-15+20\\-2-4+5&3+16-15&5+20-20\\2+3-4&-3-12+12&-5-15+16\end{bmatrix}$
$\Rightarrow\text{A}^2-\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$
$\therefore\ \text{A}^2=\text{A}$

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