Question

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

• A. $\beta=-8$
• B. $\beta=8$
• C. $a=4$
• D. $\alpha=1$

Answer 1

Answer: (A)

$ A ^2=3 A +\alpha I$
$ A ^3=3 A ^2+\alpha A$
$ A ^3=3(3 A +\alpha I )+\alpha A$
$ A ^3=9 A +\alpha A +3 \alpha I$
$ A ^4=(9+\alpha) A ^2+3 \alpha A $
$=(9+\alpha)(3 A +\alpha I )+3 \alpha A$
$ = A (27+6 \alpha)+\alpha(9+\alpha) $
$ \Rightarrow 27+6 \alpha=21 \Rightarrow \alpha=-1 $
$ \Rightarrow \beta=\alpha(9+\alpha)=-8$