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

(a) $\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right]$


$$\begin{gathered} \mathrm{Here}, \\ A=\left[\begin{array}{ccc}2& 0& -3\\ 4& 3& 1\\ -5& 7& 2\end{array}\right] \\ \Rightarrow A^T=\left[\begin{array}{ccc}2& 4& -5\\ 0& 3& 7\\ -3& 1& 2\end{array}\right] \\ \mathrm{Now}, \\ A+A^T=\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 A+A^T=\left[\begin{array}{ccc}2+2& 0+4& -3-5\\ 4+0& 3+3& 1+7\\ -5-3& 7+1& 2+2\end{array}\right] \\ \Rightarrow A+A^T=\left[\begin{array}{ccc}4& 4& -8\\ 4& 6& 8\\ -8& 8& 4\end{array}\right] \\ A-A^T=\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 A-A^T=\left[\begin{array}{ccc}2-2& 0-4& -3+5\\ 4-0& 3-3& 1-7\\ -5+3& 7-1& 2-2\end{array}\right] \\ \Rightarrow A-A^T=\left[\begin{array}{ccc}0& -4& 2\\ 4& 0& -6\\ -2& 6& 0\end{array}\right] \\ \mathrm{Let}P=\frac{1}{2}\left(A+A^T\right)=\frac{1}{2}\left[\begin{array}{ccc}4& 4& -8\\ 4& 6& 8\\ -8& 8& 4\end{array}\right]=\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right] \\ Q=\frac{1}{2}\left(A-A^T\right)=\frac{1}{2}\left[\begin{array}{ccc}0& -4& 2\\ 4& 0& -6\\ -2& 6& 0\end{array}\right]=\left[\begin{array}{ccc}0& -2& 1\\ 2& 0& -3\\ -1& 3& 0\end{array}\right] \\ \mathrm{Now}, \\ {P}^T={\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right]}^T=\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right]=P \\ {Q}^T={\left[\begin{array}{ccc}0& -2& 1\\ 2& 0& -3\\ -1& 3& 0\end{array}\right]}^T=\left[\begin{array}{ccc}0& 2& -1\\ -2& 0& 3\\ 1& -3& 0\end{array}\right]=-\left[\begin{array}{ccc}0& -2& 1\\ 2& 0& -3\\ -1& 3& 0\end{array}\right]=-Q \\ \mathrm{Thus},\mathrm{P}\mathrm{is}\mathrm{symmetric}\mathrm{and}\mathrm{Q}\mathrm{is}\mathrm{skew}-\mathrm{symmetric}. \\ P+Q=\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right]+\left[\begin{array}{ccc}0& -2& 1\\ 2& 0& -3\\ -1& 3& 0\end{array}\right] \\ =\left[\begin{array}{ccc}2+0& 2-2& -4+1\\ 2+2& 3+0& 4-3\\ -4-1& 4+3& 2+0\end{array}\right] \\ =\left[\begin{array}{ccc}2& 0& -3\\ 4& 3& 1\\ -5& 7& 2\end{array}\right]=A \\ \mathrm{Thus},\mathrm{we}\mathrm{have}\mathrm{expressed}A\mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{a}\mathrm{symmetric}\mathrm{and}\mathrm{a}\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix}. \\ \mathrm{Hence},\mathrm{the}\mathrm{symmetric}\mathrm{matrix}\mathrm{is}\left[\begin{array}{ccc}2& 2& -4\\ 2& 3& 4\\ -4& 4& 2\end{array}\right]. \end{gathered}$$