Question

If \[A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\] , show that A − A^T is a skewsymmetric matrix.

 

 

Answer 1

\[Given: \hspace{0.167em} A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\] 
\[ A^T = \begin{bmatrix}3 & 1 \\ - 4 & - 1\end{bmatrix}\] 

\[Now, \] 
\[A - A^T = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix} - \begin{bmatrix}3 & 1 \\ - 4 & - 1\end{bmatrix}\] 

\[ \Rightarrow A - A^T = \begin{bmatrix}3 - 3 & - 4 - 1 \\ 1 + 4 & - 1 + 1\end{bmatrix}\] 
\[ \Rightarrow A - A^T = \begin{bmatrix}0 & - 5 \\ 5 & 0\end{bmatrix} . . . \left( 1 \right)\] 

\[ \left( A - A^T \right)^T = \begin{bmatrix}0 & - 5 \\ 5 & 0\end{bmatrix}^T \] 
\[ \Rightarrow \left( A - A^T \right)^T = \begin{bmatrix}0 & 5 \\ - 5 & 0\end{bmatrix}\] 
\[ \Rightarrow \left( A - A^T \right)^T = - \begin{bmatrix}0 & - 5 \\ 5 & 0\end{bmatrix}\] \[ \Rightarrow \left( A - A^T \right)^T = - \left( A - A^T \right) \left[ \text{From eq .} \left( 1 \right) \right]\] 

Thus, `(A -A^T)`    is a skew  symmetric matrix.