A) \[7{{A}^{8}}\]
B) \[7{{A}^{7}}\]
C) \[8I\]
D) \[6I\]
E) \[I\]
Correct Answer: A
Solution :
\[A=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right]\] \[{{A}^{2}}=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right]\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right]\] \[{{A}^{2}}=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]={{I}_{3}}\] ?.. (i) \[{{A}^{4}}={{A}^{2}}.{{A}^{2}}={{I}_{3}}.{{I}_{3}}={{({{I}_{3}})}^{2}}={{I}_{3}}\] \[{{A}^{6}}={{A}^{2}}.{{A}^{4}}={{I}_{3}}.{{I}_{3}}={{({{I}_{3}})}^{2}}={{I}_{3}}\] Now, \[{{A}^{2}}+2{{A}^{4}}+4{{A}^{6}}\] \[={{I}_{3}}+2{{I}_{3}}+4{{I}_{3}}\] \[=7{{I}_{3}}+7{{({{I}_{3}})}^{4}}\] \[=7{{({{A}^{2}})}^{4}}=7{{A}^{8}}\] [From Eq. (i)]
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