Quadratic matrix/Rank/Invertible/Linearly independent/Fact

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

1. ${}M$ is invertible.
2. The rank of ${}M$ is ${}n$.
3. The rows of ${}M$ are linearly independent.
4. The columns of ${}M$ are linearly independent.

Proof, Write another proof