Let [] and [] be real numbers.

Question

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

Accepted Answer

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

$A^{3}=3 A^{2}+\alpha A$

$\mathrm{A}^{3}=3(3 \mathrm{~A}+\alpha \mathrm{I})+\alpha \mathrm{A}$

$\mathrm{A}^{3}=9 \mathrm{~A}+\alpha \mathrm{A}+3 \alpha \mathrm{I}$

$\mathrm{A}^{4}=(9+\alpha) \mathrm{A}^{2}+3 \alpha \mathrm{A}$

$=(9+\alpha)(3 \mathrm{~A}+\alpha \mathrm{I})+3 \alpha \mathrm{A}$

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

$\Rightarrow 27+6 \alpha=21 \Rightarrow \alpha=-1$

$\Rightarrow \beta=\alpha(9+\alpha)=-8$

Solutions · 1