Suppose that $A$ is a square matrix of size $n$. The following are equivalent.

1. $A$ is nonsingular.
2. $A$ row-reduces to the identity matrix.
3. The null space of $A$ contains only the zero vector, $\nsp{A}=\set{\zerovector}$.
4. The linear system $\linearsystem{A}{\vect{b}}$ has a unique solution for every possible choice of $\vect{b}$.
5. The columns of $A$ are a linearly independent set.
6. $A$ is invertible.
7. The column space of $A$ is $\complex{n}$, $\csp{A}=\complex{n}$.
8. The columns of $A$ are a basis for $\complex{n}$.
9. The rank of $A$ is $n$, $\rank{A}=n$.
10. The nullity of $A$ is zero, $\nullity{A}=0$.
11. The determinant of $A$ is nonzero, $\detname{A}\neq 0$.
12. $\lambda=0$ is not an eigenvalue of $A$.
13. The linear transformation $\ltdefn{T}{\complex{n}}{\complex{n}}$ defined by $\lt{T}{\vect{x}}=A\vect{x}$ is invertible.