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

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Q. 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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Solution:

$ AB =0 \Rightarrow| AB |=0 $
image
If $|A| \neq 0, B=0$ (not possible)
If $| B | \neq 0, A =0$ (not possible)
Hence $| A |=| B |=0$
$\Rightarrow AX =0$ has infinitely many solutions