Question

If A = $\left[\begin{array}{ccc}3&0&0\\0&3&0\\0&0&3\end{array}\right],$ then find A⁴

Answer 1

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Answer 2

Answer:

$A^{4}=81\:I$

Step-by-step explanation:

if

$A^{4} =A^{2}\times A^{2} \\ \\A^{2}=\left[\begin{array}{ccc}3&0&0\\0&3&0\\0&0&3\end{array}\right]\times\left[\begin{array}{ccc}3&0&0\\0&3&0\\0&0&3\end{array}\right]\\\\\\A^{2}=\left[\begin{array}{ccc}9&0&0\\0&9&0\\0&0&9\end{array}\right]$

so for

$A^{4} =A^{2}\times A^{2} \\\\=\left[\begin{array}{ccc}9&0&0\\0&9&0\\0&0&9\end{array}\right] \times \left[\begin{array}{ccc}9&0&0\\0&9&0\\0&0&9\end{array}\right]\\\\\\ =\left[\begin{array}{ccc}81&0&0\\0&81&0\\0&0&81\end{array}\right]\\\\\\=81 \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]\\\\\\A^{4}=81\:I$