A be the set of all square matrices of order 3 with elements either 0,1 , or -1 , then:

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For each place we have 3 possibilities hence $n(A)=3^3$ $ \ \left(\begin{array}{lll} a & d & e \ d & b & f \ e & f & c \end{array}ight) 	ext { for } a+b+c=0, a, b, c \in\{0,1,-1\} $ Let $a=b=c=0 \Rightarrow 3^3$ such matrices Let $a=1, b=-1, c=0 \Rightarrow 3^3$ such matrices $\Rightarrow 6 	imes 3^3$ such matrices (by permuting $\left.\mathrm{a}, \mathrm{b}, \mathrm{c}ight) \Rightarrow 7 	imes 3^3$ matrices with trace zero $ \left(\begin{array}{ccc} 0 & d & e \ -d & 0 & f \ -e & -f & 0 \end{array}ight) 	ext { skew symmetric matrices }=3^3 $ Number of matrices in A such that each of $0,1,-1$ occurs at least once $$ =3^9-{ }^3 C_1 	imes 2^9+{ }^3 C_2 	imes 1^9=19683-3 	imes 512+3=18150 $$ Skew symmetric matrix of odd order has determinant zero.