If A is a square matrix, then \((\mathrm{A}^T - \mathrm{A})\) is-
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → a skew-symmetric matrix
A matrix B is skew-symmetric if \(B^T = -B\). Let \(B = A^T - A\). Using \((A^T)^T = A\):
\(B^T = (A^T - A)^T = (A^T)^T - A^T\)
\(= A - A^T = -(A^T - A) = -B\)
So \(A^T - A\) is skew-symmetric for every square matrix A.
It is the null matrix only when A is symmetric, so Option (1) does not hold in general.