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If $M$ and $N$ are square matrices of order 3 , then which one of the following statements is not true?
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Verified Answer
The correct answer is:
For all symmetric matrices $M$ and $N$, matrix $M N$ is symmetric
For two square matrices $M$ and $N$ of order 3 ,
the matrix $M N-N M$ is skew symmetric, if $M$ and $N$ are symmetric matrices, because
$\begin{aligned} & (M N-N M)^T=(M N)^T-(N M)^T \\ & =N M-M N=-(M N-N M)\end{aligned}$
The matrix $N^T M N$ is symmetric or skew symmetric according as $M$ is symmetric or skew symmetric, because $\left(N^T M N\right)^T=N^T M^T\left(N^T\right)^T=N^T M^T N$ $= \begin{cases}N^T M N, & \text { if } M \text { is symmetric matrix } \\ -N^T M N, & \text { if } M \text { is skew symmetric matrix }\end{cases}$
The matrix $M N$ is not symmetric matrix for all symmetric matrices $M$ and $N$.
And for any two matrices $M$ and $N, \operatorname{adj}(M N)$ and $\operatorname{adj}(N M)$ need not be equal.
Hence, option (c) is correct.
the matrix $M N-N M$ is skew symmetric, if $M$ and $N$ are symmetric matrices, because
$\begin{aligned} & (M N-N M)^T=(M N)^T-(N M)^T \\ & =N M-M N=-(M N-N M)\end{aligned}$
The matrix $N^T M N$ is symmetric or skew symmetric according as $M$ is symmetric or skew symmetric, because $\left(N^T M N\right)^T=N^T M^T\left(N^T\right)^T=N^T M^T N$ $= \begin{cases}N^T M N, & \text { if } M \text { is symmetric matrix } \\ -N^T M N, & \text { if } M \text { is skew symmetric matrix }\end{cases}$
The matrix $M N$ is not symmetric matrix for all symmetric matrices $M$ and $N$.
And for any two matrices $M$ and $N, \operatorname{adj}(M N)$ and $\operatorname{adj}(N M)$ need not be equal.
Hence, option (c) is correct.
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