JEE Main 2025 (Online) 28th January Morning ShiftQuestion 1
Let $M$ denote the set of all real matrices of order $3\times 3$ and let $S=\{-3,-2,-1,1,2\}$. Let
$S_1=\{A=[a_{ij}]\in M: A=A^T\text{ and }a_{ij}\in S,\ \forall\, i,j\}$,
$S_2=\{A=[a_{ij}]\in M: A=-A^T\text{ and }a_{ij}\in S,\ \forall\, i,j\}$,
$S_3=\{A=[a_{ij}]\in M: a_{11}+a_{22}+a_{33}=0\text{ and }a_{ij}\in S,\ \forall\, i,j\}$.
If $n(S_1\cup S_2\cup S_3)=125\alpha$, then $\alpha$ equals ____.
Matrices
Answer$1613$
Solution
$S_2=\emptyset$ (diagonal would be $0\notin S$). $|S_1|=5^6=15625$. Diagonal triples from $S$ with sum $0$: $12$, so $|S_3|=12\cdot5^6=187500$ and $|S_1\cap S_3|=12\cdot5^3=1500$. Union $=15625+187500-1500=201625=125\cdot1613$.