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Let $F$ be a field, let $n$ be a positive integer, and let $A,B \in M_{n×n}(F )$ be matrices satisfying $B \neq O$ and $AB = O$. Show that $|A| = 0$.

I am stuck on this problem, any hint would be greatly appreciated.

I know $adj(AB)=adj(A)\,adj(B)$, $adj(A)\,A=A\,adj(A)=|A|\,I$, and $adj(A^T)=adj(A)^T$.

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$|A| \ne 0$ if and only if $A$ is invertible. Now, assume this is the case, and that $AB = 0$. Then $B = A^{-1}AB = A^{-1}0 = 0$.

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