Verified Solution • 2025–26 Session moderate

QJEEM23Mae3d7: JEE Main 2023 Mathematics - Class 12 JEEM23Mae3d7 Solution: Let α and β be real numbers. Consider a 3 × 3 matrix A such…

Let α and β be real numbers. Consider a 3 × 3 matrix A such that A2=3A+αI. If A4=21A+βI, then

Multiple Choice Options

  • A α=1
  • B α=4
  • C β=8
  • D β=−8

Step-by-Step Verified Solution

A2=3 A+αI\mathrm{A}^{2}=3 \mathrm{~A}+\alpha \mathrm{I}

A3=3A2+αAA^{3}=3 A^{2}+\alpha A

A3=3(3 A+αI)+αA\mathrm{A}^{3}=3(3 \mathrm{~A}+\alpha \mathrm{I})+\alpha \mathrm{A}

A3=9 A+αA+3αI\mathrm{A}^{3}=9 \mathrm{~A}+\alpha \mathrm{A}+3 \alpha \mathrm{I}

A4=(9+α)A2+3αA\mathrm{A}^{4}=(9+\alpha) \mathrm{A}^{2}+3 \alpha \mathrm{A}

=(9+α)(3 A+αI)+3αA=(9+\alpha)(3 \mathrm{~A}+\alpha \mathrm{I})+3 \alpha \mathrm{A}

=A(27+6α)+α(9+α)=\mathrm{A}(27+6 \alpha)+\alpha(9+\alpha)

⇒27+6α=21⇒α=−1\Rightarrow 27+6 \alpha=21 \Rightarrow \alpha=-1

⇒β=α(9+α)=−8\Rightarrow \beta=\alpha(9+\alpha)=-8

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