If the matrix \( \mathrm{A}=\begin{bmatrix}0 & -1 & 3x\\1 & y & -5\\-6 & 5 & 0\end{bmatrix} \) is skew-symmetric, then the value of \( (5x-y) \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 10
A skew-symmetric matrix satisfies \( a_{ji}=-a_{ij} \) for all i, j; in particular every diagonal entry is 0.
\( \displaystyle a_{22}=y=0 \)
\( \displaystyle a_{13}=-a_{31} \)
\( \displaystyle 3x=-(-6)=6 \)
\( \displaystyle x=2 \)
(The other pairs already agree: \( a_{12}=-1=-a_{21} \) and \( a_{23}=-5=-a_{32} \).)
\( \displaystyle 5x-y=5(2)-0=10 \)
Why the other options are incorrect
Option (2) \( -10 \) — Incorrect. This takes \( 3x=a_{31}=-6 \), i.e. x = −2, which is the condition for a symmetric matrix.
Option (3) 5 — Incorrect. The value is 10, not 5.
Option (4) \( -5 \) — Incorrect. The value is 10, not −5.