$\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \
a_{21} & a_{22} & a_{23} \
a_{31} & a_{32} & a_{33}
\end{array}\right]No.ofelementsin\mathrm{S}1: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \Rightarrow 5^3 \times 5^3No.ofelementsinA=-A^T \Rightarrow 0sinceno.zeroin5No.ofelementsin\mathrm{S}3 \Rightarrow\left.\begin{array}{c}a{11}+a{22}+a_{33}=0 \Rightarrow(1,2,-3) \Rightarrow 31 \ \text { or } \ (1,1,-2) \Rightarrow 3 \ \text { or } \ (-1,-1,2) \Rightarrow 3\end{array}\right} \Rightarrow 12 \times 5^6\begin{aligned} & \mathrm{n}\left(\mathrm{S}_1 \cap \mathrm{S}_3\right)=12 \times 5^3 \ & \mathrm{n}\left(\mathrm{S}_1 \cup \mathrm{S}_2 \cup \mathrm{~S}_3\right)=5^6(1+12)-12 \times 5^3 \ & \quad \Rightarrow 5^3 \times\left[13 \times 5^3-12\right]=125 \alpha \ & \quad \alpha=1613\end{aligned}$