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  4. If A = [1& 1 &3 5&2&6 -2&-1&-3] , then A is

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Q. If $ A = \begin{bmatrix}1& 1 &3\\ 5&2&6\\ -2&-1&-3\end{bmatrix} $, then A is

Matrices

Solution:

Here , $A^2 = AA $
$ = \begin{bmatrix}1&1&3\\ 5&2&6\\ -2&-1&-3\end{bmatrix} \begin{bmatrix}1&1&3\\ 5&2&6\\ -2&-1&-3\end{bmatrix}$
$ = \begin{bmatrix}0&0&0\\ 3&3&9\\ -1&-1&-3\end{bmatrix}$
Hence, $ A^{3} = A^{2} A$
$ = \begin{bmatrix}0&0&0\\ 3&3&9\\ -1&-1&-3\end{bmatrix} \begin{bmatrix}1&1&3\\ 5&2&6\\ -2&-1&-3\end{bmatrix} $
$ =\begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix}=O$ ,
$\Rightarrow $ A is a nilpotent matrix