JEE Main 2018 (Online) 15th April Evening ShiftQuestion 1
Suppose $A$ is any $3\times3$ non-singular matrix and $(A-3I)(A-5I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+\beta A^{-1}=4I$, then $\alpha+\beta$ is equal to
Matrices
Solution
$A^2-8A+15I=0$, multiply by $A^{-1}$: $A+15A^{-1}=8I$, so $\frac12A+\frac{15}2A^{-1}=4I$. $\alpha+\beta=\frac12+\frac{15}2=8$.