JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Matrices

Since $AB=0$ (zero matrix), taking determinants: $|AB|=|A||B|=0$.

Since both A and B are non-zero matrices, if $|A|\ne 0$, then B would be zero (contradiction). Similarly, if $|B|\ne 0$, then A would be zero (contradiction).

Therefore, $|A|=0$, meaning A is singular. For a singular matrix A, the homogeneous system $AX=0$ has infinitely many solutions (non-trivial solutions exist).