Question

If the matrix $\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right]=A+B$, where $A$ is symmetric and $B$ is skew-symmetric then $B=$.

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

Answer 1

The correct option is B $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

$\left[\begin{array}{cc}2& 5\\ 3& -1\end{array}\right]=A+B$

since $A$ is symmetric is of the form

$A=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$

Since $B$ is skew-symmetric is of the form

$B=\left[\begin{array}{cc}0& d\\ -d& 0\end{array}\right]$

$A+B=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]+\left[\begin{array}{cc}0& d\\ -d& 0\end{array}\right]=\left[\begin{array}{cc}a& b+d\\ b-d& c\end{array}\right]=\left[\begin{array}{cc}2& 5\\ 3& -1\end{array}\right]$

$⟹a=2,c=-1,b+d=5,b-d=3$

$⟹b=4,d=1$

So $B=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$