1. $A$ is invertible.
2. The reduced row echelon form of $A$ is $I_n$.
3. The rank of $A$ equals $n$.
4. The span of the columns of $A$ is $R_n$.
5. The equation $Ax = b$ is consistent for every $b$ in $R_n$.
6. The nullity of $A$ equals zero.
7. The columns of $A$ are linearly independent.
8. The only solution of $Ax = 0$ is $\mathbf{0}$.
9. There exists an $n \times n$ matrix $B$ such that $BA = I_n$.
10. There exists an $n \times n$ matrix $C$ such that $AC = I_n$.
11. $A$ is a product of elementary matrices.