For some , R, let A=[array|c|c| & 2 1 & 2 array] and B=[array|c|c| 1 & 1 1 & array] be suc…

Question

For some $\alpha ,\beta \in R$, let $A=[\begin{array}{|c|c|} \alpha  & 2 \ 1 & 2 \end{array}]$ and $B=[\begin{array}{|c|c|} 1 & 1 \ 1 & \beta \end{array}]$ be such that ${A}^{2}-4A+2I={B}^{2}-3B+I=O$. Then ${(\det (adj({A}^{3}-{B}^{3})))}^{2}$ is equal to $____$ .

Answer: 225

Accepted Answer

$\begin{gathered} \begin{gathered} \quad |\begin{array}{cc}\alpha -\lambda \quad 2 \ 1 \quad 2-\lambda \end{array}|=0 \ (\lambda -2)(\lambda -\alpha )-2=0 \ {\lambda }^{2}+\lambda (-2-\alpha )+2\alpha -2=0 \ \Rightarrow {A}^{2}+A(-2-\alpha )+(2\alpha -2)I=0 \ \Rightarrow -2-\alpha =-4\Rightarrow \alpha =2 \end{gathered}$

$\begin{gathered} \quad |\begin{array}{cc}1-K \quad 2 \ 1 \quad \beta -K\end{array}|=0 \ \Rightarrow (K-\beta )(K-1)-1=0 \ {K}^{2}+K(-1-\beta )+\beta -1=0 \ \Rightarrow -1-\beta =-3\Rightarrow \beta =2 \end{gathered}$

$\begin{gathered} \Rightarrow A=[2 \quad 2 \ 1 \quad 2],B=[1 \quad 1 \ 1 \quad 2] \ {A}^{3}-{B}^{3}=[15 \quad 20 \ 6 \quad 7 \end{gathered}]=C,\det (C)=-15 \ |{C|}^{2}=|C{|}^{2}=225 \end{gathered}$

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