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?

Answer 1

Since, A is a skew symmetric matrix.

∴ A^T = −A
$$\begin{gathered} {\left[\begin{array}{ccc}0& 1& -2\\ -1& 0& 3\\ x& -3& 0\end{array}\right]}^T=-\left[\begin{array}{ccc}0& 1& -2\\ -1& 0& 3\\ x& -3& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}0& -1& x\\ 1& 0& -3\\ -2& 3& 0\end{array}\right]=\left[\begin{array}{ccc}0& -1& 2\\ 1& 0& -3\\ -x& 3& 0\end{array}\right] \\ \mathrm{Corresponding}\mathrm{elements}\mathrm{of}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \Rightarrow x=2 \end{gathered}$$

Hence, the value of x is 2.