Let S=m Z:A^m^2+A^m=3I-A^-6, where A=[array|c|c| 2 & -1 1 & 0 array].

Question

Let $S={m\in Z:{A}^{{m}^{2}}+{A}^{m}=3I-{A}^{-6}}$, where $A=[\begin{array}{|c|c|} 2 & -1 \ 1 & 0 \end{array}]$. Then $n(S)$ is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 2

Accepted Answer

$\begin{gathered} \begin{gathered} \quad A=[2 \quad -1 \ 1 \quad 0] \ {A}^{2}=[3 \quad -2 \ 2 \quad -1],{A}^{3}=[4 \quad -3 \ 3 \quad -2],{A}^{4}=[5 \quad -4 \ 4 \quad -3] \ \quad {A}^{6}=[7 \quad -6 \ 6 \quad -5] \ {A}^{m}=[m+1 \quad -m \ m \quad -m+1], \ {A}^{{m}^{2}}=[{m}^{2}+1 \quad -{m}^{2} \ {m}^{2} \quad -({m}^{2}-1)] \ {A}^{{m}^{2}}+{A}^{m}=3I-{A}^{-6} \ [{m}^{2}+1 \quad -{m}^{2} \ {m}^{2} \quad -({m}^{2}-1)]+[m+1 \quad -m \ m \quad -m+1] \end{gathered}$

$\begin{gathered} =3[1 \quad 0 \ 0 \quad 1]-[-5 \quad 6 \ -6 \quad 7] \ =[8 \quad -6 \ 6 \quad -4 \end{gathered}] \ ={m}^{2}+1+m+1=8 \ ={m}^{2}+m-6=0\Rightarrow m=-3,2 \ n(s)=2 \end{gathered}$

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