If \(A=\begin{bmatrix}0&x^2-6&-3\\-x&0&-8\\x^2-2x&8&0\end{bmatrix}\) is a skew symmetric matrix, then the value(s) of \(x\) is/are-
(A) 3
(B) -3
(C) -2
(D) -1
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) only
In a skew-symmetric matrix, \(a_{ij} = -a_{ji}\) for every pair of positions. The diagonal is already zero, and \(a_{23} = -8 = -a_{32}\) holds for every x. The other two pairs give
\(a_{12} = -a_{21}\): \(x^2 - 6 = x\)
\(x^2 - x - 6 = 0\)
\((x - 3)(x + 2) = 0\), so \(x = 3\) or \(x = -2\)
\(a_{31} = -a_{13}\): \(x^2 - 2x = 3\)
\(x^2 - 2x - 3 = 0\)
\((x - 3)(x + 1) = 0\), so \(x = 3\) or \(x = -1\)
So both conditions hold together only for x = 3.
Checking the statements
(A) 3 — Correct. \(x^2 - 6 = 3 = x\) and \(x^2 - 2x = 3\).
(B) -3 — Incorrect. \(a_{12} = 3\) but \(-a_{21} = -3\).
(C) -2 — Incorrect. \(a_{31} = 4 + 4 = 8\), not 3.
(D) -1 — Incorrect. \(a_{12} = -5\) but \(-a_{21} = -1\).