Question

prove that for A=[2 4, 5 6],A+A^T is a symmetric and a skew symmetric matrix where A^T is the transpose of A.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{A}=\left[\begin{array}{cc}2& 4\\ 5& 6\end{array}\right] \\ {\mathrm{A}}^{\mathrm{T}}=\left[\begin{array}{cc}2& 5\\ 4& 6\end{array}\right] \\ \mathrm{Let}\mathrm{P}=\mathrm{A}+{\mathrm{A}}^{\mathrm{T}}=\left[\begin{array}{cc}2& 4\\ 5& 6\end{array}\right]+\left[\begin{array}{cc}2& 5\\ 4& 6\end{array}\right]=\left[\begin{array}{cc}4& 9\\ 9& 12\end{array}\right] \\ \mathrm{now},{\mathrm{P}}^{\mathrm{T}}=\left[\begin{array}{cc}4& 9\\ 9& 12\end{array}\right]=\mathrm{P} \\ \Rightarrow \mathrm{P}\mathrm{or}\mathrm{A}+{\mathrm{A}}^{\mathrm{T}}\mathrm{is}\mathrm{a}\mathrm{symmetric}\mathrm{matrix}. \\ \mathrm{Let}\mathrm{Q}=\mathrm{A}-{\mathrm{A}}^{\mathrm{T}}=\left[\begin{array}{cc}2& 4\\ 5& 6\end{array}\right]-\left[\begin{array}{cc}2& 5\\ 4& 6\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right] \\ {\mathrm{Q}}^{\mathrm{T}}=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]=-\mathrm{Q} \\ \Rightarrow \mathrm{Q}\mathrm{or}\mathrm{A}-{\mathrm{A}}^{\mathrm{T}}\mathrm{is}\mathrm{a}\mathrm{skew}\mathrm{symmetric}\mathrm{matrix}. \end{gathered}$$