A) \[7{{A}^{8}}\]
B) \[7{{A}^{7}}\]
C) 8I
D) 6I
Correct Answer: A
Solution :
[a] \[{{A}^{2}}=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right]\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]\] \[{{A}^{2}}={{A}^{4}}={{A}^{6}}={{I}_{3}}\Rightarrow {{A}^{2}}+2{{A}^{4}}+4{{A}^{6}}\] \[=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]+\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{matrix} \right]+\left[ \begin{matrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} 7 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 7 \\ \end{matrix} \right]=7{{I}_{3}}=7{{A}^{8}}\]
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