A) \[{{A}^{4}}-I={{A}^{2}}+I\]
B) \[{{A}^{3}}+I=A({{A}^{3}}-I)\]
C) \[{{A}^{3}}-I=A(A-I)\]
D) \[{{A}^{2}}+I=A({{A}^{2}}-I)\]
Correct Answer: D
Solution :
Given that \[A=\left[ \begin{matrix} 0 & -1 \\ 1 & 2 \\ \end{matrix} \right]\] \[{{A}^{2}}=\left[ \begin{matrix} -1 & 0 \\ 0 & -1 \\ \end{matrix} \right]\Rightarrow {{A}^{2}}=-I\] \[{{A}^{3}}=\left[ \begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix} \right]\] \[{{A}^{4}}=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]=I\] \[{{A}^{2}}+I={{A}^{3}}-A\] \[-I+I={{A}^{3}}-A\] \[{{A}^{3}}\ne A\]
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