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

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The system of linear equations $A X = O$ has a unique solution

$A d j (A)$ is an invertible matrix

The system of linear equations $A X = O$ has infinitely many solutions

B is an invertible matrix

answer is B.

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Detailed Solution

$A B = 0 \Rightarrow | A B | = 0$
$\begin{array}{l} | A | | B | = 0 \\ | A | = 0 o r | B | = 0 o r | A | = 0 a n d | B | = 0 \end{array}$
If $| A | \neq 0, t h e n A^{- 1} e x i s t, A^{- 1} A B = O \Rightarrow B = O (C o n t r a d i c t i o n)$
$| B | \neq 0, t h e n B^{- 1} e x i s t, A B B^{- 1} = O \Rightarrow A = O (C o n t r a d i c t i o n)$
Hence $| A | = | B | = 0$
$\Rightarrow A X = O$ has infinitely many solutions.