If A is a skew-symmetric matrix, then which of the following statemen…

The correct answer is Option (4) → (B) and (D) only

For a skew-symmetric matrix, \(A' = -A\). Each statement is checked below; the statements marked Incorrect are the ones that are NOT true.

(A) A is singular if order of A is odd — Correct. For order n,

    \(|A| = |A'| = |-A| = (-1)^n|A|\)

For odd n this gives \(|A| = -|A|\), so \(|A| = 0\).

(B) A is non-singular — Incorrect. Every skew-symmetric matrix of odd order is singular (\(|A| = 0\)); the zero matrix is another example.

(C) \(A^{2025}\) is a skew-symmetric matrix — Correct. \((A^{2025})' = (A')^{2025} = (-A)^{2025} = -A^{2025}\).

(D) \(A^{2025}\) is a symmetric matrix — Incorrect. \((A^{2025})' = -A^{2025}\), so \(A^{2025}\) is skew-symmetric and it is symmetric only when \(A^{2025} = O\). For example, take

    \(\displaystyle A = \begin{bmatrix}0&1\\-1&0\end{bmatrix},\quad A^2 = -I\)

Then \(A^{2025} = (A^2)^{1012}A = A\), which is not symmetric.

(E) all diagonal elements of A are zeros — Correct. \(a_{ii} = -a_{ii}\) gives \(a_{ii} = 0\).

So the statements that are NOT true are (B) and (D).