question_answer

If \[A=\left[ \begin{align}   & 2\,\,\,\,\,0\,\,\,\,\,0 \\  & 0\,\,\,\,\,2\,\,\,\,\,0 \\  & 0\,\,\,\,\,0\,\,\,\,\,2 \\ \end{align} \right]\],then \[{{\mathbf{A}}^{\mathbf{5}}}\]is equal to:

A)  \[16A\]                        

B)  15A          

C)  32A  

D)  8A

Correct Answer: C

Solution :

 \[A=\left[ \begin{align}   & 2\,\,\,\,0\,\,\,\,0 \\  & 0\,\,\,\,2\,\,\,\,0 \\  & 0\,\,\,\,0\,\,\,\,2 \\ \end{align} \right]\]             \[A=2.\left[ \begin{align}   & 1\,\,\,\,0\,\,\,\,0 \\  & 0\,\,\,\,1\,\,\,\,0 \\  & 0\,\,\,\,0\,\,\,\,1 \\ \end{align} \right]\]             Similarly,             \[{{A}^{5}}=32.\left[ \begin{align}   & 1\,\,\,\,0\,\,\,\,0 \\  & 0\,\,\,\,1\,\,\,\,0 \\  & 0\,\,\,\,0\,\,\,\,1 \\ \end{align} \right]=32A\] Hence option [c] is correct