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  4. If A= [ 3 3 3 3 3 3 3 3 3 ] , then A4 is equal to

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Q. If $ A= \begin{bmatrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{bmatrix} , $ then $ {{A}^{4}} $ is equal to

KEAMKEAM 2008Matrices

Solution:

Given, $ A=\left[ \begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{matrix} \right] $
$ \therefore $ $ A=3\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right| $
$ \therefore $ $ {{A}^{2}}=3\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right].3\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right] $
$=9\left[ \begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{matrix} \right]=9A $
$ \therefore $ $ {{A}^{4}}={{A}^{2}}.{{A}^{2}} $
$=9A.9A=81{{A}^{2}}=81.9A $
$=729A $