Invertible matrix simple problem

Question

I am a bit confused here, on the following question.

Determine whether this statement is true:

If matrix A is invertible and B≠0, then AB≠0

The book says it is true. But I can think about a counterexample:

If B has a row of zeros, it satisfies B≠0, but on the other hand: det(B)=0

Thus: det(AB)=det(A) ∙ det(B)=0

So what? The conclusion, at least as you have written it, is NOT that "det(AB) is not 0" but that AB is not 0. — George Ivey, Aug 06 '22 at 12:18

score 1 · Accepted Answer · answered Aug 06 '22 at 12:18

1

The statement says $AB \neq 0$, which is much weaker than saying that $\det(AB) \neq 0$. If $AB = 0$ and $A$ is invertible, we can apply $A^{-1}$ to get $B = 0$, which contradicts $B \neq 0$.

answered Aug 06 '22 at 12:18

Klaus

10,578

Thank you! Now I understand it – Zhu Bajie Aug 06 '22 at 12:24
If $AB=0$ and $A$ is invertible, multiply with the inverse of $A$ from the left. Then $B=0$ follows – Wuestenfux Aug 06 '22 at 15:47

Invertible matrix simple problem

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