Question

The matrix $A=\left[\begin{array}{ccc}0& -5& 8\\ 5& 0& 12\\ -8& -12& 0\end{array}\right]$ is a

(a) diagonal matrix
(b) symmetric matrix
(c) skew-symmetric matrix
(d) scalar matrix

Answer 1

Given: $A=\left[\begin{array}{ccc}0& -5& 8\\ 5& 0& 12\\ -8& -12& 0\end{array}\right]$
$$\begin{gathered} A^T={\left[\begin{array}{ccc}0& -5& 8\\ 5& 0& 12\\ -8& -12& 0\end{array}\right]}^T \\ =\left[\begin{array}{ccc}0& 5& -8\\ -5& 0& -12\\ 8& 12& 0\end{array}\right] \\ =-1\left[\begin{array}{ccc}0& -5& 8\\ 5& 0& 12\\ -8& -12& 0\end{array}\right] \\ =-A \end{gathered}$$

Therefore, matrix A is skew-symmetric matrix.

Hence, the correct option is (c).