JEE Advanced · Mathematics · 18. Matrices
Let \(M\) and \(N\) be two \(3 \times 3\) non-singular skew-symmetric matrices such that \(M N=N M\). If \(P^T\) denotes the transpose of \(P\), then \(M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T\) is equal to
- A
\(M^2\)
- B
\(-N^2\)
- C
\(-M^2\)
- D
\(M N\)
Answer & Solution
Correct Answer
(C)
\(-M^2\)
Step-by-step Solution
Detailed explanation
Given, \(M^T=-M, N^T=-N\)
and \(\quad M N=N M\)
\[
\begin{aligned}
\therefore & M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T \\
& =M^2 N^2 N^{-1}\left(M^T\right)^{-1}\left(N^{-1}\right)^T \cdot M^T \\
& =M^2 N\left(N N^{-1}\right)(-M)^{-1}\left(N^T\right)^{-1}(-M) \\
& =M^2 N\left(-M^{-1}\right)(-N)^{-1}(-M) \\
& =-M^2 N M^{-1} N^{-1} M \\
& =-M \cdot(M N) M^{-1} N^{-1} M \\
& =-M(N M) M^{-1} N^{-1} M \\
& =-M N\left(N M^{-1}\right) N^{-1} M \\
& =-M\left(N N^{-1}\right) M=-M^2
\end{aligned}
\]
Note Here, non-singular word should not be used, since there is no non-singular \(3 \times 3\) skew-symmetric matrix.
and \(\quad M N=N M\)
\[
\begin{aligned}
\therefore & M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T \\
& =M^2 N^2 N^{-1}\left(M^T\right)^{-1}\left(N^{-1}\right)^T \cdot M^T \\
& =M^2 N\left(N N^{-1}\right)(-M)^{-1}\left(N^T\right)^{-1}(-M) \\
& =M^2 N\left(-M^{-1}\right)(-N)^{-1}(-M) \\
& =-M^2 N M^{-1} N^{-1} M \\
& =-M \cdot(M N) M^{-1} N^{-1} M \\
& =-M(N M) M^{-1} N^{-1} M \\
& =-M N\left(N M^{-1}\right) N^{-1} M \\
& =-M\left(N N^{-1}\right) M=-M^2
\end{aligned}
\]
Note Here, non-singular word should not be used, since there is no non-singular \(3 \times 3\) skew-symmetric matrix.
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