If A is a square matrix, then $(A^T - A)$ is-

a skew-symmetric matrix

The correct answer is Option (4) → a skew-symmetric matrix

Given: $A$ is a square matrix

Expression: $A^T - A$

Note:

• If $A^T = A$, then $A$ is symmetric ⟹ $A^T - A = 0$
• If $A^T = -A$, then $A$ is skew-symmetric ⟹ $A^T - A = -2A$

But in general, for any square matrix $A$, the matrix $A^T - A$ satisfies:

$(A^T - A)^T = A - A^T = - (A^T - A)$

⟹ $A^T - A$ is a skew-symmetric matrix