Question

A square matrix $A$ of order $3$ satisfies $A^{2}=Ι-2A$ , where $Ι$ is an identity matrix of order $3$ . If $A^{n}=29A-12Ι$ , then the value of $n$ is equal to

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (C)

$A^{3}=A^{2}A=\left(Ι - 2 A\right)A=A-2A^{2}=A-2\left(Ι - 2 A\right)=5A-2Ι$
$A^{4}=A^{3}A=\left(5 A - 2 Ι\right)A=5A^{2}-2A=5\left(Ι - 2 A\right)-2A=5Ι-12A$
$A^{5}=A^{4}\left(A\right)=\left(5 Ι - 12 A\right)A=5A-12A^{2}=5A-12\left(Ι - 2 A\right)=29A-12Ι$