JEE Advance - Mathematics (2011 - Paper 1 Offline - No. 19)
Let M and N be two 3 $$\times$$ 3 non-singular skew symmetric matrices such that MN = NM. If PT denotes the transpose of P, then M2N2(MTN)$$-$$1(MN$$-$$1)T is equal to
M2
$$-$$N2
$$-$$M2
MN
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Given, $${M^T} = - M$$, $${N^T} = - N$$
and $$MN = NM$$ ..... (i)
$$\therefore$$ $${M^2}{N^2}{({M^T}N)^{ - 1}}{(M{N^{ - 1}})^T}$$
$$ = {M^2}{N^2}{N^{ - 1}}{({M^T})^{ - 1}}{({N^{ - 1}})^T}.{M^T}$$
$$ = {M^2}N(N{M^{ - 1}}){( - M)^{ - 1}}{({N^T})^{ - 1}}( - M)$$
$$ = {M^2}NI( - {M^{ - 1}}){( - N)^{ - 1}}( - M)$$
$$ = - {M^2}N{M^{ - 1}}{N^{ - 1}}M$$
$$ = - M.(MN){M^{ - 1}}{N^{ - 1}}M$$
$$ = - M(NM){M^{ - 1}}{N^{ - 1}}M$$
$$ = - MN(N{M^{ - 1}}){N^{ - 1}}M$$
$$ = - M(N{N^{ - 1}})M = - {M^2}$$
Note : This question is wrong, as given. An odd order skew symmetric matrix can't be invertible. Had the matrix be of even order, it could have been correct.