Let $AB \in K^{n\times n}$, $AB$ is a diagonal matrix and the elements on the diagonal are non-zero.
Is $A$ invertible?
Since $AB$ is a diagonal matrix and the elements on the diagonal are non-zero, $AB$ is invertible. Also neither $A$ nor $B$ can possibly contain zero-vectors or $AB$ would too which it doesn't. However not containing zero-vectors is not sufficient for a matrix to be invertible, right?
Please be aware that I don't know much about linear algebra yet, so a thorough explanation would be much appreciated.