Question

Express the matrix $A=\left[\begin{array}{ccc}4& 2& -1\\ 3& 5& 7\\ 1& -2& 1\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[\begin{array}{ccc}4& 2& -1\\ 3& 5& 7\\ 1& -2& 1\end{array}\right] \\ {A}^T=\left[\begin{array}{ccc}4& 3& 1\\ 2& 5& -2\\ -1& 7& 1\end{array}\right] \\ \mathrm{Let}X=\frac{1}{2}\left(A+A^T\right)=\frac{1}{2}\left(\left[\begin{array}{ccc}4& 2& -1\\ 3& 5& 7\\ 1& -2& 1\end{array}\right]+\left[\begin{array}{ccc}4& 3& 1\\ 2& 5& -2\\ -1& 7& 1\end{array}\right]\right)=\left[\begin{array}{ccc}4& \frac{5}{2}& 0\\ \frac{5}{2}& 5& \frac{5}{2}\\ 0& \frac{5}{2}& 1\end{array}\right] \\ {X}^T={\left[\begin{array}{ccc}4& \frac{5}{2}& 0\\ \frac{5}{2}& 5& \frac{5}{2}\\ 0& \frac{5}{2}& 1\end{array}\right]}^T=\left[\begin{array}{ccc}4& \frac{5}{2}& 0\\ \frac{5}{2}& 5& \frac{5}{2}\\ 0& \frac{5}{2}& 1\end{array}\right]=X \\ \mathrm{Let}Y=\frac{1}{2}\left(A-A^T\right)=\frac{1}{2}\left(\left[\begin{array}{ccc}4& 2& -1\\ 3& 5& 7\\ 1& -2& 1\end{array}\right]-\left[\begin{array}{ccc}4& 3& 1\\ 2& 5& -2\\ -1& 7& 1\end{array}\right]\right)=\left[\begin{array}{ccc}0& \frac{-1}{2}& -1\\ \frac{1}{2}& 0& \frac{9}{2}\\ 1& \frac{-9}{2}& 0\end{array}\right] \\ {Y}^T={\left[\begin{array}{ccc}0& \frac{-1}{2}& -1\\ \frac{1}{2}& 0& \frac{9}{2}\\ 1& \frac{-9}{2}& 0\end{array}\right]}^T=\left[\begin{array}{ccc}0& \frac{1}{2}& 1\\ \frac{-1}{2}& 0& \frac{-9}{2}\\ -1& \frac{9}{2}& 0\end{array}\right]=-\left[\begin{array}{ccc}0& \frac{-1}{2}& -1\\ \frac{1}{2}& 0& \frac{9}{2}\\ 1& \frac{-9}{2}& 0\end{array}\right]=-Y \\ \mathrm{Thus},X\mathrm{is}\mathrm{a}\mathrm{symmetric}\mathrm{matrix}\mathrm{and}Y\mathrm{is}\mathrm{a}\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix}. \\ \mathrm{Now}, \\ X+Y=\left[\begin{array}{ccc}4& \frac{5}{2}& 0\\ \frac{5}{2}& 5& \frac{5}{2}\\ 0& \frac{5}{2}& 1\end{array}\right]+\left[\begin{array}{ccc}0& \frac{-1}{2}& -1\\ \frac{1}{2}& 0& \frac{9}{2}\\ 1& \frac{-9}{2}& 0\end{array}\right]=\left[\begin{array}{ccc}4& 2& -1\\ 3& 5& 7\\ 1& -2& 1\end{array}\right]=A \end{gathered}$$