IIT-JEE 2011 · Paper 1 · Mathematics Q55

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MathematicsMultiple correct+4Question 55 of 69

Let $M$ and $N$ be two $3\times3$ non-singular skew-symmetric matrices such that $MN=NM$ . If $P^T$ denotes the transpose of $P$ , then $M^2N^2\left(M^TN\right)^{-1}\left(MN^{-1}\right)^T$ is equal to

M^2

-N^2

-M^2

MN

Answer and worked solution

Official answer: Bonus: full marks to everyone

Official key: marks to all.

Why. A $3\times3$ skew-symmetric matrix satisfies $\det M=\det M^T=\det(-M)=-\det M$ , so $\det M=0$ : no non-singular $M$ or $N$ exists, and the expression is undefined.

Formal working (ignoring this): with $M^T=-M$ , $N^T=-N$ and $MN=NM$ ,
$(M^TN)^{-1}=-N^{-1}M^{-1},\qquad(MN^{-1})^T=(N^{-1})^TM^T=N^{-1}M,$
so $M^2N^2(M^TN)^{-1}(MN^{-1})^T=-M^2N^2N^{-1}M^{-1}N^{-1}M=-M^2$ , which would give (C).

Topics:Special matrices Adjoint and inverse

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IIT-JEE 2011 · Paper 1, Mathematics question 55