Question

If $A=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$, prove that A − A^T is a skew-symmetric matrix.

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}A=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right] \\ {A}^T=\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right] \\ \mathrm{Now}, \\ \left(A-A^T\right)=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]-\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right] \\ \Rightarrow \left(A-A^T\right)=\left[\begin{array}{cc}2-2& 3-4\\ 4-3& 5-5\end{array}\right] \\ \Rightarrow \left(A-A^T\right)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]...\left(1\right) \\ {\left(A-A^T\right)}^T={\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]}^T \\ \Rightarrow {\left(A-A^T\right)}^T=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right] \\ \Rightarrow {\left(A-A^T\right)}^T=-\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right] \\ \Rightarrow \left(A-A^T\right)=-{\left(A-A^T\right)}^T\left[\mathrm{Using}\mathrm{eq}.\left(1\right)\right] \\ \mathrm{Thus},\left(A-A^T\right)\mathrm{is}\mathrm{a}\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix}. \end{gathered}$$