A) A is a zero matrix
B) \[A=(-1),I\]where I is a unit matrix
C) \[{{A}^{-1}}\]does not exist
D) \[{{A}^{2}}=I\]
Correct Answer: D
Solution :
The given matrix\[A=\left[ \begin{matrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \\ \end{matrix} \right]\] (a) It is clear that A is not a zero matrix. (b)\[(-1)I=-1\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ \end{matrix} \right]\ne A\] ie., \[(-1)I\ne A\] (c)\[|A|=0\left| \begin{matrix} -1 & 0 \\ 0 & 0 \\ \end{matrix} \right|-0\left| \begin{matrix} 0 & 0 \\ -1 & 0 \\ \end{matrix} \right|-1\left| \begin{matrix} 0 & -1 \\ -1 & 0 \\ \end{matrix} \right|\] \[=0-0-1(-1)=1\] Since,\[|A|\ne 0\].So\[{{A}^{-1}}\]exists. (d) \[{{A}^{2}}=A.A\] \[=\left[ \begin{matrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \\ \end{matrix} \right]\left[ \begin{matrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \\ \end{matrix} \right]\] \[\Rightarrow \] \[{{A}^{2}}=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]\] \[\Rightarrow \] \[{{A}^{2}}=I\]
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