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Choose the correct answer in the following questions:
If $\mathbf{A}, \mathbf{B}$ are symmetric matrices of same order then $\mathbf{A B}-\mathbf{B A}$ is $\mathbf{a}$
(a) Skew - symmetric matrix
(b) Symmetric matrix
(c) Zeromatrix
(d) Identity matrix
If $\mathbf{A}, \mathbf{B}$ are symmetric matrices of same order then $\mathbf{A B}-\mathbf{B A}$ is $\mathbf{a}$
(a) Skew - symmetric matrix
(b) Symmetric matrix
(c) Zeromatrix
(d) Identity matrix
Solution:
1438 Upvotes
Verified Answer
Now $\mathrm{A}^{\prime}=\mathrm{B}, \mathrm{B}^{\prime}=\mathrm{B}$
$$
\begin{aligned}
(\mathrm{AB}-\mathrm{BA})^{\prime} &=(\mathrm{AB})^{\prime}-(\mathrm{BA})^{\prime} \\
&=\mathrm{B}^{\prime} \mathrm{A}^{\prime}-\mathrm{A}^{\prime} \mathrm{B}^{\prime}=\mathrm{BA}-\mathrm{AB} \\
&=-(\mathrm{AB}-\mathrm{BA})
\end{aligned}
$$
$\mathrm{AB}-\mathrm{BA}$ is a skew-symmetric matrix
Hence, option (a) is correct.
$$
\begin{aligned}
(\mathrm{AB}-\mathrm{BA})^{\prime} &=(\mathrm{AB})^{\prime}-(\mathrm{BA})^{\prime} \\
&=\mathrm{B}^{\prime} \mathrm{A}^{\prime}-\mathrm{A}^{\prime} \mathrm{B}^{\prime}=\mathrm{BA}-\mathrm{AB} \\
&=-(\mathrm{AB}-\mathrm{BA})
\end{aligned}
$$
$\mathrm{AB}-\mathrm{BA}$ is a skew-symmetric matrix
Hence, option (a) is correct.
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