A) \[{{A}^{3}}+3{{A}^{2}}-I=O\]
B) \[{{A}^{3}}-3{{A}^{2}}-I=O\]
C) \[{{A}^{3}}+2{{A}^{2}}-I=O\]
D) \[{{A}^{3}}-{{A}^{2}}+I=O\]
Correct Answer: B
Solution :
\[{{A}^{2}}=AA=\left[ \,\begin{matrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \\ \end{matrix}\, \right]\,\left[ \,\begin{matrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \\ \end{matrix}\, \right]\]= \[\left[ \,\begin{matrix} 2 & 3 & 1 \\ 5 & 6 & 2 \\ 3 & 4 & 1 \\ \end{matrix}\, \right]\] Þ\[{{A}^{3}}={{A}^{2}}A=\left[ \,\begin{matrix} 2 & 3 & 1 \\ 5 & 6 & 2 \\ 3 & 4 & 1 \\ \end{matrix}\, \right]\,\,\left[ \,\begin{matrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \\ \end{matrix}\, \right]\]=\[\left[ \,\begin{matrix} 7 & 9 & 3 \\ 15 & 19 & 6 \\ 9 & 12 & 4 \\ \end{matrix}\, \right]\] Here\[{{A}^{3}}-3{{A}^{2}}=\left[ \,\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix}\, \right]\,=I\]Þ \[{{A}^{3}}-3{{A}^{2}}-I=0\].
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