Let , be real numbers.

Question

Let $\alpha,\beta$ be real numbers. Consider a $3	imes3$ matrix $A$ such that $A^2=3A+\alpha I$. If $A^4=21A+\beta I$, then:

Accepted Answer

$A^3=3A^2+\alpha A=3(3A+\alpha I)+\alpha A=(9+\alpha)A+3\alpha I$.

$A^4=(9+\alpha)A^2+3\alpha A=(9+\alpha)(3A+\alpha I)+3\alpha A=A(27+6\alpha)+\alpha(9+\alpha)I$.

Comparing with $A^4=21A+\beta I$: $27+6\alpha=21\Rightarrow\alpha=-1$, and $\beta=\alpha(9+\alpha)=(-1)(8)=-8$.

Solutions · 1