Question

The matrix $\left[\begin{array}{ccc}0& 5& -7\\ -5& 0& 11\\ 7& -11& 0\end{array}\right]$ is
(a) a skew-symmetric matrix
(b) a symmetric matrix
(c) a diagonal matrix
(d) an uppertriangular matrix

Answer 1

(a) a skew-symmetric matrix


Here,

A = $\left[\begin{array}{ccc}0& 5& -7\\ -5& 0& 11\\ 7& -11& 0\end{array}\right]$

$\Rightarrow$A^T = $\left[\begin{array}{ccc}0& -5& 7\\ 5& 0& -11\\ -7& 11& 0\end{array}\right]$

$$\begin{gathered} \Rightarrow A^T=-\left[\begin{array}{ccc}0& 5& -7\\ -5& 0& 11\\ 7& -11& 0\end{array}\right] \\ \Rightarrow A^T=-A \end{gathered}$$


Thus, A is a skew-symmetric matrix.