Question

If matrix $A=\left[\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]$, can be written as $B+C$ where B is symmetric matrix and C is skew-symmetric matrix, then $B-C$ is equal to

• A. $\left[\begin{array}{ccc}0& 3& 4\\ -3& 0& -7\\ -4& 7& 0\end{array}\right]$
• B. $\left[\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]$
• C. $\left[\begin{array}{ccc}0& -3& -4\\ 3& 0& 7\\ 4& -7& 0\end{array}\right]$
• D. $\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]$

Here matrix A is expressed as the sum of symmetric and skew-symmetric matrix. Then
$B=\frac{1}{2}(A+A^T)$ and $C=\frac{1}{2}(A-A^T)$
Now
$A=\left[\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]\Rightarrow A^T=\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]$
$\Rightarrow B=\frac{1}{2}(\left[\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]+\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right])=\frac{1}{2}\left[\begin{array}{ccc}0& 5& 2\\ 5& -6& 1\\ 2& 1& 8\end{array}\right]$
and
$C=\frac{1}{2}(\left[\begin{array}{ccc}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]-\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right])=\frac{1}{2}\left[\begin{array}{ccc}0& -3& -4\\ 3& 0& 7\\ 4& -7& 0\end{array}\right]$
$\therefore B-C=\left[\begin{array}{ccc}0& 4& 3\\ 1& -3& -3\\ -1& 4& 4\end{array}\right]$