JEE MAIN - Mathematics (2004 - No. 13)
statement about the matrix $$A$$ is
$${A^2} = 1$$
$$A=(-1)I,$$ where $$I$$ is a unit matrix
$${A^{ - 1}}$$ does not exist
$$A$$ is a zero matrix
Explanation
$$A = \left[ {\matrix{
0 & 0 & { - 1} \cr
0 & { - 1} & 0 \cr
{ - 1} & 0 & 0 \cr
} } \right]$$
clearly $$\,\,\,A \ne 0.\,$$ Also $$\,\,\left| A \right| = - 1 \ne 0$$
$$\therefore$$ $${A^{ - 1}}\,\,$$ exists, further
$$\left( { - 1} \right)I = \left[ {\matrix{ { - 1} & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right] \ne A$$
Also $${A^2} = \left[ {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right]\left[ {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right]$$
$$ = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right] = I$$
clearly $$\,\,\,A \ne 0.\,$$ Also $$\,\,\left| A \right| = - 1 \ne 0$$
$$\therefore$$ $${A^{ - 1}}\,\,$$ exists, further
$$\left( { - 1} \right)I = \left[ {\matrix{ { - 1} & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right] \ne A$$
Also $${A^2} = \left[ {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right]\left[ {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right]$$
$$ = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right] = I$$
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