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 \( \mathbf{A} \).