Question

If $A=\left[\begin{array}{ccc}2& 0& -3\\ 4& 3& 1\\ -5& 7& 2\end{array}\right]$ is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is

• A. $\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right]$
• B. $\left[\begin{array}{ccc}2& 4& -5\\ 0& 3& 7\\ -3& 1& 2\end{array}\right]$
• C. $\left[\begin{array}{ccc}4& 4& -8\\ 4& 6& 8\\ -8& 8& 4\end{array}\right]$
• D. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right]$

Let $A=P+Q$
where $P=\frac{1}{2}(A+A^T)$ is symmetric and $Q=\frac{1}{2}(A-A^T)$ is skew-symmetric.
Given, $A=\left[\begin{array}{ccc}2& 0& -3\\ 4& 3& 1\\ -5& 7& 2\end{array}\right]$
$A^T=\left[\begin{array}{ccc}2& 4& -5\\ 0& 3& 7\\ -3& 1& 2\end{array}\right]$
So, $P=\frac{1}{2}(\left[\begin{array}{ccc}2& 0& -3\\ 4& 3& 1\\ -5& 7& 2\end{array}\right]+\left[\begin{array}{ccc}2& 4& -5\\ 0& 3& 7\\ -3& 1& 2\end{array}\right])$
$\Rightarrow P=\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right]$