Which of the following matrices is a skew-symmetric matrix?

The correct answer is Option (3) → \(\begin{bmatrix}0&2&-3\\-2&0&1\\3&-1&0\end{bmatrix}\)

A square matrix is skew-symmetric when \(a_{ji} = -a_{ij}\) for all i and j. Taking i = j gives \(a_{ii} = -a_{ii}\), so every diagonal element must be zero.

For the matrix in Option (3), the diagonal elements are all 0 and

    \(a_{12} = 2,\quad a_{21} = -2\)
    \(a_{13} = -3,\quad a_{31} = 3\)
    \(a_{23} = 1,\quad a_{32} = -1\)

The two elements of each pair placed symmetrically about the diagonal are negatives of each other, so the matrix is skew-symmetric.

Why the other options are incorrect

• Option (1) \(\begin{bmatrix}1&2\\-2&1\end{bmatrix}\) — Incorrect. Its diagonal elements are 1, not 0.

• Option (2) \(\begin{bmatrix}0&2&3\\2&0&1\\3&1&0\end{bmatrix}\) — Incorrect. Here \(a_{ij} = a_{ji}\), so it is symmetric, not skew-symmetric.

• Option (4) \(\begin{bmatrix}0&-2&3\\2&0&-1\\3&1&0\end{bmatrix}\) — Incorrect. \(a_{13} = 3\) and \(a_{31} = 3\), so \(a_{31} \ne -a_{13}\).