Corollaries from Rank-nullity

For a square matrix $A\in M_{n\times n}(\mathbb R)$, the following statements are equivalent:

1. $Ax=u$ has a unique solution for every $u\in\mathbb R^n$;
2. $Ax=0$ has only the trivial solution;
3. every column of the reduced echelon form of $A$ is leading;
4. $\det(A)\ne0$;
5. $A$ is invertible;
6. the columns of $A$ are linearly independent;
7. the columns of $A$ span $\mathbb R^n$;
8. the columns of $A$ form a basis for $\mathbb R^n$;
9. $\operatorname{rank}(A)=n$;
10. $\operatorname{nullity}(A)=0$.