For any square matrix A with real number entries, consider the fo… - Vidyadip

MCQ

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.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
✓
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

✓

Answer

Correct option: B.

Both (A) and (R) are true but (R) is not the correct explanation of (A).

(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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