Question
If $\text{A}=\begin{bmatrix}0&0\\4&0\end{bmatrix},$ find A16.
✓
Answer
Given,
$\text{A}=\begin{bmatrix}0&0\\4&0\end{bmatrix}$
$ \text{A}^2=\text{A}\times\text{A}$
$ =\begin{bmatrix}0&0\\4&0\end{bmatrix}\begin{bmatrix}0&0\\4&0\end{bmatrix}$
$ =\begin{bmatrix}0+0&0+0\\0+0&0+0\end{bmatrix}$
$=\begin{bmatrix}0&0\\0&0\end{bmatrix}$
$=0$
$ \text{A}^4=\text{A}^2\times\text{A}^2$
$=0\times0$
$=0$
$ \text{A}^{16}=\text{A}^4\times\text{A}^4$
$=0\times0$
$=0$
So,
A16 is a null matrix
$\text{A}=\begin{bmatrix}0&0\\4&0\end{bmatrix}$
$ \text{A}^2=\text{A}\times\text{A}$
$ =\begin{bmatrix}0&0\\4&0\end{bmatrix}\begin{bmatrix}0&0\\4&0\end{bmatrix}$
$ =\begin{bmatrix}0+0&0+0\\0+0&0+0\end{bmatrix}$
$=\begin{bmatrix}0&0\\0&0\end{bmatrix}$
$=0$
$ \text{A}^4=\text{A}^2\times\text{A}^2$
$=0\times0$
$=0$
$ \text{A}^{16}=\text{A}^4\times\text{A}^4$
$=0\times0$
$=0$
So,
A16 is a null matrix
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