Let A be an n x n - matrix.
Then the following statements are equivalent:
- A is an invertible matrix.
- A is row equivalent to I_{n}.
- A has n pivot positions.
- The equation Ax=0 has only the
trivial solution.
- The columns of A form a linearly independent set.
- The linear transformation
T:R^{n}→R^{n};
T(x)=Ax is one-to-one.
- The equation Ax = b has at least one
solution for each b in R^{n}.
- The columns of A span R^{n}.
- The linear transformation
T:R^{n}→R^{n};
T(x)=Ax is onto.
- There is an n x n - matrix C
such that CA=I_{n}.
- There is an n x n - matrix D
such that AD=I_{n}.
- A^{T} is an invertible matrix.