If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.
Solution
\[Given: \hspace{0.167em} A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix} \]
\[ A^T = \begin{bmatrix}2 & 4 \\ 3 & 5\end{bmatrix}\]
\[Now, \]
\[\left( A - A^T \right) = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix} - \begin{bmatrix}2 & 4 \\ 3 & 5\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = \begin{bmatrix}2 - 2 & 3 - 4 \\ 4 - 3 & 5 - 5\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix} . . . \left( 1 \right)\]
\[ \left( A - A^T \right)^T = \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix}^T \]
\[ \Rightarrow \left( A - A^T \right)^T = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right)^T = - \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix}\]
\[ \Rightarrow \left( A - A^T \right) = - \left( A - A^T \right)^T \left[ \text{Using eq} . \left( 1 \right) \right]\]
\[Thus, \left( A - A^T \right) \text{is a skew - symmetric matrix} .\]
