Mathematics - Matrices Question with Solution | TestHub

$\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$ No. of elements in ${\mathrm{S}}_1:\mathrm{A}={\mathrm{A}}^{\mathrm{T}}\Rightarrow 5^3\times 5^3$ No. of elements in $A=-A^T\Rightarrow 0$ since no. zero in 5 No. of elements in ${\mathrm{S}}_3\Rightarrow$ $\begin{array}{c}{a}_{11}+a_{22}+a_{33}=0\Rightarrow (1,2,-3)\Rightarrow 31\\ \text{ or }\\ (1,1,-2)\Rightarrow 3\\ \text{ or }\\ (-1,-1,2)\Rightarrow 3\end{array}\}\Rightarrow 12\times 5^6$ $\begin{array}{rl}& \mathrm{n}\left({\mathrm{S}}_1\cap {\mathrm{S}}_3\right)=12\times 5^3\\ & \mathrm{n}\left({\mathrm{S}}_1\cup {\mathrm{S}}_2\cup {\mathrm{S}}_3\right)=5^6(1+12)-12\times 5^3\\ & \Rightarrow 5^3\times \left[13\times 5^3-12\right]=125\alpha \\ & \alpha =1613\end{array}$