A) \[{{A}^{3}}+3{{A}^{2}}-I=0\]
B) \[{{A}^{3}}-3{{A}^{2}}-I=0\]
C) \[{{A}^{3}}+2{{A}^{2}}-I=0\]
D) \[{{A}^{3}}-{{A}^{2}}+I=0\]
E) \[{{A}^{3}}+{{A}^{2}}-I=0\]
Correct Answer: B
Solution :
\[\because \]\[A=\left[ \begin{matrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 0 \\ \end{matrix} \right]\] \[\therefore \]\[{{A}^{2}}=\left[ \begin{matrix} 2 & 3 & 1 \\ 5 & 6 & 2 \\ 3 & 4 & 1 \\ \end{matrix} \right]\]and\[{{A}^{3}}=\left[ \begin{matrix} 7 & 9 & 3 \\ 15 & 19 & 6 \\ 9 & 12 & 4 \\ \end{matrix} \right]\] Hence, \[{{A}^{3}}-3{{A}^{2}}-I=0\]
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