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Matrices and Determinants question

2025 · 28 Jan · Shift 1 · Q46

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Matrices and Determinants
2025 · 28 Jan · Shift 1 · Q46

Matrices and Determinants question

2025 · 28 Jan · Shift 1 · Q46

JEE MainMathematicsMatrices and DeterminantsNumerical+4 / −1

Let M denote the set of all real matrices of order

3 \times 3

and let

\mathrm{S}=\{-3,-2,-1,1,2\}

. Let

\begin{aligned} & \mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \\ & \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \\ & \mathrm{S}_3=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0 \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\} . \end{aligned}

If

n\left(S_1 \cup S_2 \cup S_3\right)=125 \alpha

, then

\alpha

equls

\underline{\hspace{2cm}}

.

Numerical answer

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