JEE Main 2024 (Online) 9th April Evening ShiftQuestion 1
Let $B=\begin{bmatrix}1&3\\1&5\end{bmatrix}$ and $A$ be a $2\times2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^{4}+\alpha C^{2}+\beta I=O$, then $2\beta-\alpha$ is equal to
Matrices
Solution
$AB^{-1}=A^{-1}\Rightarrow AB^{-1}A=I\Rightarrow B^{-1}=A^{-2}\Rightarrow A^2=B$. $C=B^{-1}AB$ is similar to $A$, so $C^4+\alpha C^2+\beta I=O\iff A^4+\alpha A^2+\beta I=O\iff B^2+\alpha B+\beta I=O$. Cayley-Hamilton for $B$: $B^2-6B+2I=O$ (tr $=6$, det $=5-3=2$). So $\alpha=-6,\beta=2$. $2\beta-\alpha=4+6=10$. Option (C).