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_{i j}] \in M: A=A^T \text{ and } a_{i j} \in S, \forall i, j\} , S_2= \{A= [a_{i j}] \in M: A=-A^T \text{ and } a_{i j} \in S, \forall i, j\} , S_3= \{A= [a_{i j}] \in M: a_{11}+a_{22}+a_{33}=0 \text{ and } a_{i j} \in S,. \forall i, j \} . If n (S_1 \cup S_2 \cup S_3) =125 \alpha, then \alpha equals ______ [28 Jan 2025 Shift 1]

JEE Main 28 Jan 2025 Shift 1 Paper

Section: Mathematics

Question : 22 of 75

Marks: +1, -0

Let M denote the set of all real matrices of order 3×3 and let S={−3,−2,−1,1,2}. Let
‌S1={A=[aij]∈M:A=AT‌ and ‌aij∈S,∀i,j},
‌S2={A=[aij]∈M:A=−AT‌ and ‌aij∈S,∀i,j},
‌S3={A=[aij]∈M:a11+a22+a33=0‌ and ‌aij∈S,.‌∀i,j}.

If n(S1∪S2∪S3)=125α, then α equals ______

[28 Jan 2025 Shift 1]

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a11	a12	a13
a21	a22	a23
a31	a32	a33

a11+a22+a33=0	⇒(1,‌2,‌−3)⇒31
‌ or ‌(1,‌1,‌−2)	⇒3
‌ or ‌(−1,‌−1,‌2)	⇒3