Theorem. Let \(A\) be an \(n\)-by-\(n\) matrix. Then the following statements are equivalent. That is, for a given \(A\), the statements are either all true or all false.
\(A\) is an invertible matrix.
\(A\) is row equivalent to the \(n\)-by-\(n\) identity matrix.
\(A\) has \(n\) pivot positions.
The equation \(A\mathbf{x}=\mathbf{0}\) has only the trivial solution.
The columns of \(A\) form a linearly independent set.
The linear transformation \(\mathbf{x} \mapsto A\mathbf{x}\) is one-to-one.
The equation \(A\mathbf{x} = \mathbf{b}\) has at least one solution for each \(\mathbf{b} \in \mathbb{R}^n\).
The columns of \(A\) span \(\mathbb{R}^n\).
The linear transformation \(A \mapsto A\mathbf{x}\) is onto.
There is an \(n\)-by-\(n\) matrix \(C\) such that \(CA = I\).
There is an \(n\)-by-\(n\) matrix \(D\) such that \(AD = I\).
\(A^T\) is an invertible matrix.
The columns of \(A\) form a basis of \(\mathbb{R}^n\).
\(\mathrm{Col}\, A = \mathbb{R}^n\).
\(\mathrm{dim}\, \mathrm{Col}\, A = n\).
\(\mathrm{rank}\, A = n\).
\(\mathrm{Nul}\, A = \{\mathbf{0}\}\).
\(\mathrm{dim}\, \mathrm{Nul}\, A = 0\).