Question

Express the matrix $A$ as the sum of a symmetric and skew-symmetric matrix where $A=\left[\begin{array}{ccc}3& -2& -4\\ 3& -2& -5\\ -1& 1& 2\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{ccc}3& -2& -4\\ 3& -2& -5\\ -1& 1& 2\end{array}\right]$

$A^T=\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right]$

Let $P=\frac{1}{2}(A+A^T)=\frac{1}{2}(\left[\begin{array}{ccc}3& -2& -4\\ 3& -2& -5\\ -1& 1& 2\end{array}\right]+\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right])$

$=\frac{1}{2}\left(\left[\begin{array}{ccc}6& 1& -5\\ 1& -4& -4\\ -5& -4& 4\end{array}\right]\right)$

$=\left[\begin{array}{ccc}3& \frac{1}{2}& \frac{-5}{2}\\ \frac{1}{2}& -2& -2\\ \frac{-5}{2}& -2& 2\end{array}\right]$

$P^T=\left[\begin{array}{ccc}3& \frac{1}{2}& \frac{-5}{2}\\ \frac{1}{2}& -2& -2\\ \frac{-5}{2}& -2& 2\end{array}\right]=P$

Since $P=P^T$

$P$ is a symmetric matrix

$Q=\frac{1}{2}(A-A^T)=\frac{1}{2}(\left[\begin{array}{ccc}3& -2& -4\\ 3& -2& -5\\ -1& 1& 2\end{array}\right]-\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right])$

$Q=\frac{1}{2}\left(\left[\begin{array}{ccc}0& -5& -3\\ 5& 0& -6\\ 3& 6& 0\end{array}\right]\right)$

$Q=\left[\begin{array}{ccc}0& \frac{-5}{2}& \frac{-3}{2}\\ \frac{5}{2}& 0& -3\\ \frac{3}{2}& 3& 0\end{array}\right]$

$Q^T=\left[\begin{array}{ccc}0& \frac{5}{2}& \frac{3}{2}\\ \frac{-5}{2}& 0& 3\\ \frac{-3}{2}& -3& 0\end{array}\right]=-Q$

Since $Q^T=Q$

$Q$ is a skew symmetric matrix.

Now,$P+Q=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)=A$

Thus,$A$ is a sum of symmetric and skew symmetric matrix.