JEE MAIN - Mathematics (2022 - 29th July Morning Shift - No. 3)

Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then

the system of linear equations $$A X=0$$ has a unique solution

the system of linear equations $$A X=0$$ has infinitely many solutions

B is an invertible matrix

$$\operatorname{adj}(\mathrm{A})$$ is an invertible matrix

AB is zero matrix

$$ \Rightarrow |A| = |B| = 0$$

So neither A nor B is invertible

If $$|A| = 0$$

$$ \Rightarrow |\mathrm{adj}\,A| = 0$$ so $$\mathrm{adj}\,A$$

$$AX = 0$$ is homogeneous system and $$|A| = 0$$

So, it is having infinitely many solutions