For an $N\times N$ matrix $\mathbf{A}$ the following properties are all equivalent

• If the inverse of $\mathbf{A}$ exists, $\mathbf{A}$ is nonsingular.
• The equation $\mathbf{Ax}=0$ implies $\mathbf{x}=0$.
• The rows of $\mathbf{A}$ form a basis of $R^N$.
• The columns of $\mathbf{A}$ form a basis of $R^N$.
• $\mathbf{A}$ is a product of elementary matrices.
• $0$ is not eigenvalue of $\mathbf{A}$.