QJEEM25Mafe4d: JEE Main 2025 Mathematics - Class 12 JEEM25Mafe4d Solution: Let M denote the set of all real matrices of order and let .…

$\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right]$

No. of elements in $\mathrm{S}_1: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \Rightarrow 5^3 \times 5^3$

No. of elements in $S_2: A=-A^T \Rightarrow 0$

since no zero in $\mathrm{S}_2$

No. of elements in $\mathrm{S}_3 \Rightarrow$

$\left.\begin{array}{c}a_{11}+a_{22}+a_{33}=0 \Rightarrow(1,2,-3) \Rightarrow 31 \\ \text { or } \\ (1,1,-2) \Rightarrow 3 \\ \text { or } \\ (-1,-1,2) \Rightarrow 3\end{array}\right\} \Rightarrow 12 \times 5^6$

$\begin{aligned}& \mathrm{n}\left(\mathrm{~S}_1 \cap \mathrm{~S}_3\right)=12 \times 5^3 \\& \mathrm{n}\left(\mathrm{~S}_1 \cup \mathrm{~S}_2 \cup \mathrm{~S}_3\right)=5^6(1+12)-12 \times 5^3 \\& \quad \Rightarrow 5^3 \times\left[13 \times 5^3-12\right]=125 \alpha \\& \quad \alpha=1613\end{aligned}$