Question

if $A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$ is the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then $C$ is

• A. $\left[\begin{array}{cc}1& -\frac{5}{2}\\ \frac{5}{2}& 0\end{array}\right]$
• B. $\left[\begin{array}{cc}1& -\frac{5}{2}\\ \frac{5}{2}& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}0& -\frac{5}{2}\\ \frac{5}{2}& 0\end{array}\right]$
• D. $\left[\begin{array}{cc}0& -\frac{3}{2}\\ \frac{5}{2}& 1\end{array}\right]$

Answer 1

Answer: (C)

$A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$
$A=\left(\frac{A+A^{\prime }}{2}\right)+\left(\frac{A-A^{\prime }}{2}\right)=B+C$
[where $B$ and $C$ are symmetric and skew-symmetric matrices respectively]
Now, $C=\frac{A-A^{\prime }}{2}=\frac{1}{2}\left\{\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]-\left[\begin{array}{cc}3& 1\\ -4& -1\end{array}\right]\right\}$
$=\frac{1}{2}\left[\begin{array}{cc}0& -5\\ 5& 0\end{array}\right]=\left[\begin{array}{cc}0& -\frac{5}{2}\\ \frac{5}{2}& 0\end{array}\right]$