SATHEE: Matrices and Determinants 3 Question 9

Matrices and Determinants 3 Question 9

11. 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 $M^{2} N^{2} {(M^{T} N)}^{- 1} {(M N^{- 1})}^{T}$ is equal to

(a) $M^{2}$

(b) $- N^{2}$

(c) $- M^{2}$

(d) $M N$

Show Answer

Answer:

Correct Answer: 11. (c)

Solution:

Given, $M^{T} = - M, N^{T} = - N$ and $M N = N M$

$∴ M^{2} N^{2} {(M^{T} N)}^{- 1} {(M N^{- 1})}^{T}$

$\Rightarrow M^{2} N^{2} N^{- 1} {(M^{T})}^{- 1} {(N^{- 1})}^{T} \cdot M^{T}$

$\Rightarrow M^{2} N (N N^{- 1}) (- M)^{- 1} {(N^{T})}^{- 1} (- M)$

$\Rightarrow M^{2} N I (- M^{- 1}) (- N)^{- 1} (- M)$

$\Rightarrow - M^{2} N M^{- 1} N^{- 1} M$

$\Rightarrow - M \cdot (M N) M^{- 1} N^{- 1} M = - M (N M) M^{- 1} N^{- 1} M$

$\Rightarrow - M N (N M^{- 1}) N^{- 1} M = - M (N N^{- 1}) M \Rightarrow - M^{2}$

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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