Question:
For what value of $x$, is the matrix $A=\left[\begin{array}{ccc}0 & 1 & -2 \\ -1 & 0 & 3 \\ x & -3 & 0\end{array}\right]$ a skew-symmetric matrix?
Solution:
Since, A is a skew symmetric matrix.
$\therefore A^{\top}=-A$
$\left[\begin{array}{ccc}0 & 1 & -2 \\ -1 & 0 & 3 \\ x & -3 & 0\end{array}\right]^{T}=-\left[\begin{array}{ccc}0 & 1 & -2 \\ -1 & 0 & 3 \\ x & -3 & 0\end{array}\right]$
$\Rightarrow\left[\begin{array}{ccc}0 & -1 & x \\ 1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right]=\left[\begin{array}{ccc}0 & -1 & 2 \\ 1 & 0 & -3 \\ -x & 3 & 0\end{array}\right]$
Corresponding elements of equal matrices are equal.
$\Rightarrow x=2$
Hence, the value of $x$ is 2 .