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Matrices and Determinants question

2022 · 26 Jun · Shift 2 · Q42
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Matrices and Determinants question

2022 · 26 Jun · Shift 2 · Q42

JEE MainMathematicsMatrices and DeterminantsNumerical+4 / −1
Let X=[010001000], Y=αI+βX+γX2X = \left[ {\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{matrix} } \right],\,Y = \alpha I + \beta X + \gamma {X^2}X=​000​100​010​​,Y=αI+βX+γX2 and Z=α2I−αβX+(β2−αγ)X2Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma ){X^2}Z=α2I−αβX+(β2−αγ)X2, α\alphaα, β\betaβ, γ∈\gamma\inγ∈ R. If Y−1=[15−2515015−250015]{Y^{ - 1}} = \left[ {\begin{matrix} {{1 \over 5}} & {{{ - 2} \over 5}} & {{1 \over 5}} \\ 0 & {{1 \over 5}} & {{{ - 2} \over 5}} \\ 0 & 0 & {{1 \over 5}} \\ \end{matrix} } \right]Y−1=​51​00​5−2​51​0​51​5−2​51​​​, then (α−β\alpha-\betaα−β+γ\gammaγ)2 is equal to ‾\underline{\hspace{2cm}}​.
Numerical answer
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