JEE Main 2022 (Online) 29th July Morning ShiftQuestion 1
Let $A$ and $B$ be two $3\times3$ non-zero real matrices such that $AB$ is a zero matrix. Then
Matrices
Solution
$AB=O$ with $B\ne O$ means $A$ kills a nonzero column of $B$, so $A$ is singular and $AX=0$ has infinitely many solutions. Similarly $B$ is singular, and $\mathrm{adj}(A)$ is singular for rank $\le 2$ ($\det\mathrm{adj}A=(\det A)^2=0$). Option B.