A) \[\left[ \begin{matrix} {{2}^{n}} & {{2}^{n}} \\ {{2}^{n}} & {{2}^{n}} \\ \end{matrix} \right]\]
B) \[\left[ \begin{matrix} 2n & 2n \\ 2n & 2n \\ \end{matrix} \right]\]
C) \[\left[ \begin{matrix} {{2}^{2n-1}} & {{2}^{2n-1}} \\ {{2}^{2n-1}} & {{2}^{2n-1}} \\ \end{matrix} \right]\]
D) \[\left[ \begin{matrix} {{2}^{2n+1}} & {{2}^{2n+1}} \\ {{2}^{2n+1}} & {{2}^{2n+1}} \\ \end{matrix} \right]\]
Correct Answer: C
Solution :
[c] Given matrix is: \[A=\left[ \begin{matrix} 2 & 2 \\ 2 & 2 \\ \end{matrix} \right]\] \[{{A}^{2}}=\left[ \begin{matrix} 2 & 2 \\ 2 & 2 \\ \end{matrix} \right]\left[ \begin{matrix} 2 & 2 \\ 2 & 2 \\ \end{matrix} \right]=\left[ \begin{matrix} 4+4 & 4+4 \\ 4+4 & 4+4 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} {{2}^{3}} & {{2}^{3}} \\ {{2}^{3}} & {{2}^{3}} \\ \end{matrix} \right]\] \[{{A}^{3}}=\left[ \begin{matrix} 8 & 8 \\ 8 & 8 \\ \end{matrix} \right]\left[ \begin{matrix} 2 & 2 \\ 2 & 2 \\ \end{matrix} \right]=\left[ \begin{matrix} 16+16 & 16+16 \\ 16+16 & 16+16 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} 32 & 32 \\ 32 & 32 \\ \end{matrix} \right]=\left[ \begin{matrix} {{2}^{5}} & {{2}^{5}} \\ {{2}^{5}} & {{2}^{5}} \\ \end{matrix} \right]\] Going this way we get \[{{A}^{4}}=\left[ \begin{matrix} {{2}^{7}} & {{2}^{7}} \\ {{2}^{7}} & {{2}^{7}} \\ \end{matrix} \right]\] \[\Rightarrow {{A}^{n}}=\left[ \begin{matrix} {{2}^{2n-1}} & {{2}^{2n-1}} \\ {{2}^{2n-1}} & {{2}^{2n-1}} \\ \end{matrix} \right]\]
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