Determinant/Zero, linear dependent and rank property/Fact

Let ${}K$ be a field, and let ${}M$ denote an ${}n\times n$-matrix over ${}K$. Then the following statements are equivalent.

1. We have ${}\det M\neq 0$.
2. The rows of ${}M$ are linearly independent.
3. ${}M$ is invertible.
4. We have ${}\operatorname {rk} \,M=n$.

Proof, Write another proof