Q. Let M And N Be Two 3×3 Non-singular Skew-symmetric Matrices ...

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4. Let M and N be two 3 × 3 non-singular skew-symmetric matrices such that M N = NM. If P T denotes the transpose of P, then M2N2(MTN)-1(MN-1)T is equal to

Q. 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 4816 276 IIT JEEIIT JEE 2011Determinants Report Error A

M2

16% B

−N2

15% C

−M2

56% D

MN

14%

Solution:

Given, MT=−M,NT=−N and MN=NM....(i) ∴M2N2(MTN)−1(MN−1)T ⇒M2N2N−1(MT)−1(N−1)T.MT ⇒M2N(NN−1)(−M)−1(NT)−1(−M) ⇒M2NI(−M−1)(−N)−1(−M)⇒−M2NM−1N−1M ⇒−M.(MN)M−1N−1M=−M(NM)M−1N−1M ⇒−MN(NM−1)N−1M=−M(NN−1)M⇒−M2 NOTE Here, non-singular word should not be used, since there is no non-singular 3×3 skew-symmetric matrix.