QJEEM24Mab428: JEE Main 2024 Mathematics - Class 12 JEEM24Mab428 Solution: Let and be a matrix such that . If and , then is equal to

$\begin{aligned}& B=\left[\begin{array}{ll}1 & 3 \\1 & 5\end{array}\right] \\& A B^{-1}=A^{-1} \\& \Rightarrow A^2=B\end{aligned}$

Also, $B C B^{-1}=A$

$\begin{aligned}\Rightarrow C & =B^{-1} A B \\\Rightarrow C^4 & =\left(B^{-1} A B\right)\left(B^{-1} A B\right)\left(B^{-1} A B\right)\left(B^{-1} A B\right) \\& =B^{-1} A^4 B \\& =B^{-1} B^2 B \\\Rightarrow \quad C^4 & =B^2\end{aligned}$

Also, $C^2=\left(B^{-1} A B\right)\left(B^{-1} A B\right)$

$\begin{aligned}& =B^{-1} A^2 B \\& =B^{-1} B B\end{aligned}$

$\begin{aligned}& \Rightarrow C^2=B \\& \Rightarrow C^4+\alpha C^2+\beta I=0 \\& \Rightarrow B^2+\alpha B+\beta I=0 \\& B^2=\left[\begin{array}{ll}1 & 3 \\1 & 5\end{array}\right]\left[\begin{array}{ll}1 & 3 \\1 & 5\end{array}\right]=\left[\begin{array}{ll}4 & 18 \\6 & 28\end{array}\right] \\& \Rightarrow\left[\begin{array}{ll}4 & 18 \\6 & 28\end{array}\right]+\alpha\left[\begin{array}{ll}1 & 3 \\1 & 5\end{array}\right]+\left[\begin{array}{ll}\beta & 0 \\0 & \beta\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\0 & 0\end{array}\right] \\& \Rightarrow 4+\alpha+\beta=0\end{aligned}$

and $18+3 \alpha=0$

$\begin{aligned}& \Rightarrow \alpha=-6 \\& \Rightarrow \beta=2 \\& \Rightarrow 2 \beta-\alpha=10\end{aligned}$