If A= -1&0&0 0&-1&0 0&0&-1 , find A^3. - Vidyadip

Question

If $\text{A}=\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix},$ find $A^3$.

✓

Answer

Given,
$\text{A}=\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}$
$\text{A}^2=\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}$
$=\begin{bmatrix}1+0+0&0+0+0&0+0+0\\0+0+0&0+1+0&0+0+0\\0+0+0&0+0+0&0+1+0\end{bmatrix}$
$=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
$\text{A}^3=\text{A}^2\times\text{A}$
$=\begin{bmatrix}1&0&0\\1&0&0\\0&0&1\end{bmatrix}\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}$
$=\begin{bmatrix}-1+0+0&0+0+0&0+0+0\\0+0+0&0-1+0&0+0+0\\0+0+0&0+0+0&0+0-1\end{bmatrix}$
$=\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}$
$\text{A}^3=\text{A}$
Hence,
$\text{A}^3=\text{A}$

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