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Consider the following statements in respect of symmetric matrices $A$ and $B$
$1. A B$ is symmetric.
$2. A^{2}+B^{2}$ is symmetric.
Which of the above statement(s) is/are correct?
Options:
$1. A B$ is symmetric.
$2. A^{2}+B^{2}$ is symmetric.
Which of the above statement(s) is/are correct?
Solution:
1132 Upvotes
Verified Answer
The correct answer is:
2 only
We know, a matrix $\mathrm{A}$ is said to be symmetric matrix if $A^{\prime}=A$ where ' ' ' represents the transpose.
Consider $(A B)^{\prime}=B^{\prime} A^{\prime}$
Since, $(A B)^{\prime} \neq A B$
$\therefore \mathrm{AB}$ is not symmetric.
and consider $\left(A^{2}+B^{2}\right)^{\prime}=\left(A^{\prime}\right)^{2}+\left(B^{\prime}\right)^{2}=A^{2}+B^{2}$
$\therefore \mathrm{A}^{2}+\mathrm{B}^{2}$ is symmetric.
Consider $(A B)^{\prime}=B^{\prime} A^{\prime}$
Since, $(A B)^{\prime} \neq A B$
$\therefore \mathrm{AB}$ is not symmetric.
and consider $\left(A^{2}+B^{2}\right)^{\prime}=\left(A^{\prime}\right)^{2}+\left(B^{\prime}\right)^{2}=A^{2}+B^{2}$
$\therefore \mathrm{A}^{2}+\mathrm{B}^{2}$ is symmetric.
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