Let $\alpha$ and $\beta$ be real numbers. Consider a 3 $\times$ 3 matrix A such that $A^2=3A+\alpha I$. If $A^4=21A+\beta I$, then
Solution
$\mathrm{A}^{2}=3 \mathrm{~A}+\alpha \mathrm{I}$
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$A^{3}=3 A^{2}+\alpha A$
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$\mathrm{A}^{3}=3(3 \mathrm{~A}+\alpha \mathrm{I})+\alpha \mathrm{A}$
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$\mathrm{A}^{3}=9 \mathrm{~A}+\alpha \mathrm{A}+3 \alpha \mathrm{I}$
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$\mathrm{A}^{4}=(9+\alpha) \mathrm{A}^{2}+3 \alpha \mathrm{A}$
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$=(9+\alpha)(3 \mathrm{~A}+\alpha \mathrm{I})+3 \alpha \mathrm{A}$
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$=\mathrm{A}(27+6 \alpha)+\alpha(9+\alpha)$
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$\Rightarrow 27+6 \alpha=21 \Rightarrow \alpha=-1$
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$\Rightarrow \beta=\alpha(9+\alpha)=-8$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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