Matrices Question 7
Question 7 - 29 January - Shift 1
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
(1) $\alpha=1$
(2) $\alpha=4$
(3) $\beta=8$
(4) $\beta=-8$
Show Answer
Answer: (4)
Solution:
Formula: Properties Of Positive Integral Powers Of A Square Matrix, Properties of Scalar Multiplication, Properties of Matrix Multiplication
$A^{2}=3 A+\alpha I$
$A^{3}=3 A^{2}+\alpha A$
$A^{3}=3(3 A+\alpha I)+\alpha A$
$A^{3}=9 A+\alpha A+3 \alpha I$
$A^{4}=(9+\alpha) A^{2}+3 \alpha A$
$=(9+\alpha)(3 A+\alpha I)+3 \alpha A$
$=A(27+6 \alpha)+\alpha(9+\alpha)$
$\Rightarrow 27+6 \alpha=21 \Rightarrow \alpha=-1$
$\Rightarrow \beta=\alpha(9+\alpha)=-8$
