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Question

Let M and N be two 3×3 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

A
M N
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B
N2
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C
M2
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D
M2
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Solution

The correct option is B −N2Answer (Statement of the question is incorrect) Every skew-symmetric matrix of odd order is always singular and inverse of a singular matrix does not exist. The statement of the given question seems incorrect. If the matrix is given to be symmetric and non-singular, then MT=M (MN)T=MN (as MN = NM given) M2N2(MTN)−1(MN−1)T =M2N2(M N)−1((NT)−1MT) =M2N2(N−1M−1)(N−1M) =M2NM−1N−1M =M2 If there may exist a non-singular skew-symmetric matrix (which does not exist), then M2N2(MTN)−1(MN−1)T=M2N2(−MN)−1((N−1)TMT) =−M2NM−1(−N−1(−M))=−M2

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