Q.A square matrix $A$ is said to be skew-symmetric if:

Skew-symmetry is defined by the transpose equalling the negative of the matrix.

By definition, a square matrix is skew-symmetric when its transpose equals its negative, i.e. $A^{\mathsf{T}}=-A$ (equivalently $a_{ij}=-a_{ji}$, which also forces every diagonal entry to be zero). So option (b) is correct.

Checking the other options. Option (a) $A^{\mathsf{T}}=A$ is the definition of a symmetric matrix, not a skew-symmetric one. Option (c) $A=A^{-1}$ describes an involutory matrix, unrelated to skew-symmetry. Option (d) $|A|=0$ says $A$ is singular; while every odd-order skew-symmetric matrix does happen to be singular, a zero determinant is not the definition of skew-s …

The trap is confusing symmetric (A^T = A) with skew-symmetric (A^T = -A). Remember: 'skew' flips the sign, so the transpose equa …