Given a matrix A= […] .Which of the following statements is/are TRUE…

The correct answer is Option (3) → (A) and (C) only

Taking the transpose of A:

    \( \displaystyle A^T=\begin{bmatrix}0 & -3 & 8\\3 & 0 & -13\\-8 & 13 & 0\end{bmatrix} \)

    \( \displaystyle =-A \)

So A is skew-symmetric. Expanding the determinant along the first row:

    \( \displaystyle |A|=0(0+169)-3(0-104)+(-8)(39-0) \)

    \( \displaystyle =312-312 \)

    \( \displaystyle =0 \)

So \( A^T=-A \) and \( |A|=0 \).

Checking the statements

(A) The matrix A is a skew-symmetric matrix. — Correct. \( A^T=-A \).

(B) The matrix A is a symmetric matrix. — Incorrect. \( A^T=-A\ne A \), since A is not the zero matrix.

(C) The matrix A is a singular matrix. — Correct. \( |A|=0 \).

(D) The matrix A is a non-singular matrix. — Incorrect. \( |A|=0 \), so A is singular.

Tip: Every skew-symmetric matrix of odd order has determinant 0, so it is always singular; no calculation is needed.