Let B=\left[\begin{array}{ll}1 & 3 \ 1 & 5\end{array} ight] and A be a 2 imes 2 matrix such that A B^{-1}=A^{-1}. If B C B^{-1}=A and C^4+\alpha C^2+\beta I=O, then 2 \beta-\alpha is equal to

Question

Let B=\left[\begin{array}{ll}1 & 3 \ 1 & 5\end{array}ight] and A be a 2 	imes 2 matrix such that A B^{-1}=A^{-1}. If B C B^{-1}=A and C^4+\alpha C^2+\beta I=O, then 2 \beta-\alpha is equal to

Accepted Answer

Answer: B. 10 \begin{aligned} & B=\left[\begin{array}{ll} 1 & 3 \ 1 & 5 \end{array}ight] \ & A B^{-1}=A^{-1} \ & \Rightarrow A^2=B \end{aligned} Also, B C B^{-1}=A \begin{aligned} \Rightarrow C & =B^{-1} A B \ \Rightarrow C^4 & =\left(B^{-1} A Bight)\left(B^{-1} A Bight)\left(B^{-1} A Bight)\left(B^{-1} A Bight) \ & =B^{-1} A^4 B \ & =B^{-1} B^2 B \ \Rightarrow \quad C^4 & =B^2 \end{aligned} Also, C^2=\left(B^{-1} A Bight)\left(B^{-1} A Bight) \begin{aligned} & =B^{-1} A^2 B \ & =B^{-1} B B \end{aligned} \begin{aligned} & \Rightarrow C^2=B \ & \Rightarrow C^4+\alpha C^2+\beta I=0 \ & \Rightarrow B^2+\alpha B+\beta I=0 \ & B^2=\left[\begin{array}{ll} 1 & 3 \ 1 & 5 \end{array}ight]\left[\begin{array}{ll} 1 & 3 \ 1 & 5 \end{array}ight]=\left[\begin{array}{ll} 4 & 18 \ 6 & 28 \end{array}ight] \ & \Rightarrow\left[\begin{array}{ll} 4 & 18 \ 6 & 28 \end{array}ight]+\alpha\left[\begin{array}{ll} 1 & 3 \ 1 & 5 \end{array}ight]+\left[\begin{array}{ll} \beta & 0 \ 0 & \beta \end{array}ight]=\left[\begin{array}{ll} 0 & 0 \ 0 & 0 \end{array}ight] \ & \Rightarrow 4+\alpha+\beta=0 \end{aligned} and 18+3 \alpha=0 \begin{aligned} & \Rightarrow \alpha=-6 \ & \Rightarrow \beta=2 \ & \Rightarrow 2 \beta-\alpha=10 \end{aligned}

Let $$B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$$ and $$A$$ be a $$2 \times 2$$ matrix such that $$A B^{-1}=A^{-1}$$. If $$B C B^{-1}=A$$ and $$C^4+\alpha C^2+\beta I=O$$, then $$2 \beta-\alpha$$ is equal to

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