Let A and B be two 33 non-zero real matrices such that AB is the zero matrix.

Question

Let $A$ and $B$ be two $3	imes3$ non-zero real matrices such that $AB$ is the zero matrix. Then

Accepted Answer

Since $AB=O$ with $B
e O$, $A$ cannot be invertible (otherwise $B=A^{-1}\cdot O=O$, contradiction), so $|A|=0$. Similarly $|B|=0$.

With $|A|=0$, the homogeneous system $AX=0$ has non-trivial solutions, hence infinitely many solutions.

($\operatorname{adj}A$ is also singular when $|A|=0$ and $\operatorname{rank}(A)<n-1$, so (D) fails in general; (C) is false since $|B|=0$.)

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