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Matrices — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMatrices1 Mark
Question
For any square matrix $A$ with real number entries, consider the following statements.
Assertion (A) : $A+A^{\prime}$ is a symmetric matrix.
Reason (R): $A-A^{\prime}$ is a skew-symmetric matrix.
✓

Answer

(b) : Let $B=A+A^{\prime}$, then
$
B^{\prime}=\left(A+A^{\prime}\right)^{\prime}=A^{\prime}+\left(A^{\prime}\right)^{\prime}=A^{\prime}+A=A+A^{\prime}=B
$
Therefore, $B=A+A^{\prime}$ is a symmetric matrix.
Now, let $C=A-A^{\prime}$
$
C^{\prime}=\left(A-A^{\prime}\right)^{\prime}=A^{\prime}-\left(A^{\prime}\right)^{\prime}=A^{\prime}-A=-\left(A-A^{\prime}\right)=-C
$
Therefore, $C=A-A^{\prime}$ is a skew-symmetric matrix.

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