Theorem


Let A be an n x n - matrix. Then the following statements are equivalent:

  1. A is an invertible matrix.
  2. A is row equivalent to In.
  3. A has n pivot positions.
  4. The equation Ax=0 has only the trivial solution.
  5. The columns of A form a linearly independent set.
  6. The linear transformation T:RnRn; T(x)=Ax is one-to-one.
  7. The equation Ax = b has at least one solution for each b in Rn.
  8. The columns of A span Rn.
  9. The linear transformation T:RnRn; T(x)=Ax is onto.
  10. There is an n x n - matrix C such that CA=In.
  11. There is an n x n - matrix D such that AD=In.
  12. AT is an invertible matrix.