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 $0s$, three $1s$, and three $(-1)s$, then what is the number of such matrices?

Answer 1

Find the number of matrices:

Representation of a skew-symmetric matrix is of the form,

In skew symmetric matrix diagonal elements are zero.

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

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

Therefore, total matrices are $=2^3=8$.

Hence, there are $8$ such matrices