A) \[{{A}^{2}}=I\]
B) \[A=(-1)\,I,\]where I is a unit matrix
C) \[{{A}^{-1}}\]does not exist
D) A is a zero matrix
Correct Answer: A
Solution :
Let \[A=\left( \begin{matrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \\ \end{matrix} \right)\] Check by options. (i) \[{{A}^{2}}=\left( \begin{matrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \\ \end{matrix} \right)\,\,\left( \begin{matrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \\ \end{matrix} \right)\] \[{{A}^{2}}=\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right)=I\] (ii) \[(-1)\,I=\left( \begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ \end{matrix} \right)\ne A\]. (iii) \[|A|=1\ne 0\Rightarrow {{A}^{-1}}\] exists. (iv) Clearly \[A\], is not a zero matrix.
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