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.

  1. \(A\) is an invertible matrix.

  2. \(A\) is row equivalent to the \(n\)-by-\(n\) identity matrix.

  3. \(A\) has \(n\) pivot positions.

  4. The equation \(A\mathbf{x}=\mathbf{0}\) has only the trivial solution.

  5. The columns of \(A\) form a linearly independent set.

  6. The linear transformation \(\mathbf{x} \mapsto A\mathbf{x}\) is one-to-one.

  7. The equation \(A\mathbf{x} = \mathbf{b}\) has at least one solution for each \(\mathbf{b} \in \mathbb{R}^n\).

  8. The columns of \(A\) span \(\mathbb{R}^n\).

  9. The linear transformation \(A \mapsto A\mathbf{x}\) is onto.

  10. There is an \(n\)-by-\(n\) matrix \(C\) such that \(CA = I\).

  11. There is an \(n\)-by-\(n\) matrix \(D\) such that \(AD = I\).

  12. \(A^T\) is an invertible matrix.

  13. The columns of \(A\) form a basis of \(\mathbb{R}^n\).

  14. \(\mathrm{Col}\, A = \mathbb{R}^n\).

  15. \(\mathrm{dim}\, \mathrm{Col}\, A = n\).

  16. \(\mathrm{rank}\, A = n\).

  17. \(\mathrm{Nul}\, A = \{\mathbf{0}\}\).

  18. \(\mathrm{dim}\, \mathrm{Nul}\, A = 0\).