Question

Let $A$ be the set of all 3 $\times$ 3 skew-symmetric matrices whose entries are either $-1$, $0$, or $1$. If there are exactly three $0$s, three $1$s, and three $(-1)$s, then the number of such matrices is

Answer 1

Skew symmetric matrix is of form

$\left[\begin{array}{ccc}0& {\lambda }_1& {\lambda }_2\\ {-\lambda }_1& 0& {\lambda }_3\\ {-\lambda }_2& {-\lambda }_3& 0\end{array}\right]$

Here ${\lambda }_1,{\lambda }_2,{\lambda }_3\in \{1,-1\}$

$\therefore$ Total matrices $=2^3=8$

${\lambda }_1$ can be $1,-1$, ${\lambda }_2$ can be $1,-1$ and so on.