Prove that A − A^T is Skew-Symmetric

Given:

\[ A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \]

Step 1: Find A^T

\[ A^T = \begin{bmatrix} 2 & 4 \\ 3 & 5 \end{bmatrix} \]

Step 2: Compute A − A^T

\[ A – A^T = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} – \begin{bmatrix} 2 & 4 \\ 3 & 5 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]

Step 3: Find Transpose of Result

\[ (A – A^T)^T = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} = – \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]

Conclusion:

\[ (A – A^T)^T = -(A – A^T) \]

Hence, A − A^T is a skew-symmetric matrix.