A) Idempotent matrix
B) Involutory matrix
C) Nilpotent matrix
D) None of these
Correct Answer: B
Solution :
[b] \[{{A}^{2}}=\left[ \begin{matrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \\ \end{matrix} \right]\left[ \begin{matrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} 25-24+0 & 40-40+0 & 0+0+0 \\ -15+15+0 & -24+25+0 & 0+0+0 \\ -5+6-1 & -8+10-2 & 0+0+1 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]=I\] Hence, the matrix A is involutory.
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