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.