Matrices and Determinants 1 Question 12
12. For $3 \times 3$ matrices $M$ and $N$, which of the following statement(s) is/are not correct?
(2013 Adv.)
(a) $N^{T} M N$ is symmetric or skew-symmetric, according as $M$ is symmetric or skew-symmetric
(b) $M N-N M$ is symmetric for all symmetric matrices $M$ and $N$
(c) $M N$ is symmetric for all symmetric matrices $M$ and $N$
(d) $(\operatorname{adj} M)(\operatorname{adj} N)=\operatorname{adj}(M N)$ for all invertible matrices $M$ and $N$
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Answer:
Correct Answer: 12. (c,d)
Solution:
- (a) $\left(N^{T} M N\right)^{T}=N^{T} M^{T}\left(N^{T}\right)^{T}=N^{T} M^{T} N$, is symmetric if $M$ is symmetric and skew-symmetric, if $M$ is skew-symmetric.
$ \text { (b) } \begin{aligned} (M N-N M)^{T} & =(M N)^{T}-(N M)^{T} \\ & =N M-M N=-(M N-N M) \end{aligned} $
$\therefore$ Skew-symmetric, when $M$ and $N$ are symmetric.
(c) $(M N)^{T}=N^{T} M^{T}=N M \neq M N$
$\therefore$ Not correct.
(d) $(\operatorname{adj} M N)=(\operatorname{adj} N) \cdot(\operatorname{adj} M)$
$\therefore$ Not correct.
