Let M denote the set of all real matrices of order and let S = {–3, –2, –1, 1, 2}. Let , , . If , then equals __________. [2025]

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Solution

Q.
Let M denote the set of all real matrices of order $3 \times 3$ and let S = {–3, –2, –1, 1, 2}. Let

$S_{1} = {A = [a_{i j}] \in M : A = A^{T} and a_{i j} \in S, \forall i, j}$ ,

$S_{2} = {A = [a_{i j}] \in M : A = - A^{T} and a_{i j} \in S, \forall i, j}$ ,

$S_{3} = {A = [a_{i j}] \in M : a_{11} + a_{22} + a_{33} = 0 and a_{i j} \in S, \forall i, j}$ .

If $n (S_{1} \cup S_{2} \cup S_{3}) = 125 α$ , then $α$ equals __________. [2025]

Ans.
(1613)

Let M denotes the set of all real matrices of order $3 \times 3$ .

$M = {[\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]; a_{i j \in R}}$

Now, $S_{1} = {A = [a_{i j}] \in M : A = A^{T} and a_{i j} \in S, \forall i, j}$

$∴$ Number of elements in $S_{1} = 5^{6}$

$S_{2} = {A = [a_{i j}] \in M : A = - A^{T} and a_{i j} \in S, \forall i, j}$

$\Rightarrow$ Number of elements in $S_{2} = 0$

$S_{3} = {A = [a_{i j}] \in M : a_{11} + a_{22} + a_{33} = 0 and a_{i j} \in S, \forall i, j}$

$\Rightarrow$ Number of elements in $S_{3} = 12 \times 5^{6}$

[ $∵$ Possible cases are (1, 2, –3) $\to$ 3!, (1, 1, –2) $\to$ 3 and (–1, –1, 2) $\to$ 3]

$∴ n (S_{1} \cap S_{3}) = 12 \times 5^{3}$

Now, $n (S_{1} \cup S_{2} \cup S_{3}) = 5^{6} + 12 \times 5^{6} + 0 - 12 \times 5^{3}$

$= 5^{6} (1 + 12) - 12 \times 5^{3} = 5^{3} [13 \times 5^{3} - 12) = 125 α$

$\Rightarrow α = 1613$ .

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