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JEE Advanced 2011Paper 1
Exam
JEE Advanced
Year
2011
Paper
Paper 1
Subject
Mathematics
Chapter
Matrices

JEE Advanced 2011 · Paper 1 · Mathematics · Matrices

Let M and N be two 3 × 3 non-singular skew-symmetric matrices such...

Let MM and NN be two 3×33 \times 3 non-singular skew-symmetric matrices such that MN=NMM N=N M. If PTP^T denotes the transpose of PP, then M2N2M^2 N^2\left(M^T N\right)^{-1}(MN−1)T\left(M N^{-1}\right)^T is equal to

Options

  1. AM2M^2
  2. B−N2-N^2
  3. C−M2-M^2
  4. DMNM N

Correct answer

C. −M2-M^2

Step-by-step solution

Given, MT=−M,NT=−NandM^T=-M, N^T=-N and\quad M N=N M∴M2N2(MTN)−1(MN−1)T=M2N2N−1(MT)−1(N−1)T\cdotMT=M2N(NN−1)(−M)−1(NT)−1(−M)=M2N(−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\begin{aligned} \therefore& M² N²\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T \\ & =M² N² N^{-1}\left(M^T\right)^{-1}\left(N^{-1}\right)^T \cdotM^T \\ & =M² N\left(N N^{-1}\right)(-M)^{-1}\left(N^T\right)^{-1}(-M) \\ & =M² N\left(-M^{-1}\right)(-N)^{-1}(-M) \\ & =-M² N M^{-1} N^{-1} M \\ & =-M \cdot(M N) M^{-1} N^{-1} M \\ & =-M(N M) M^{-1} N^{-1} M \\ & =-M N\left(N M^{-1}\right) N^{-1} M \\ & =-M\left(N N^{-1}\right) M=-M² \end{aligned}NoteHere,non−singularwordshouldnotbeused,sincethereisnonon−singularNote Here, non-singular word should not be used, since there is no non-singular3 \times 3$ skew-symmetric matrix.
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