Question

For any square matrix $A$ with real number entries, consider the following statements
I. $A+A^{\prime}$ is a symmetric matrix.
II. $A-A^{\prime}$ is a skew-symmetric matrix.
Choose the correct option.

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. Neither I nor II is true

Answer 1

Answer: (C)

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