Characterizations of Invertible Matrices
The Invertible Matrix Theorem
Theorem: Let be a square n x n matrix. Then the following statements are equivalent:
- is an invertible matrix.
- is row equivalent to the n x n identity matrix.
- has n pivot positions.
- The equation has only the trivial solution.
- The column vectors of form a linearly independent set.
- The linear transformation is injective.
- The equation has at least one solution for each b in .
- The column vectors of span .
- The linear transformation is surjective.
- There is an n x n matrix such that .
- There is an n x n matrix such that .
- is an invertible matrix.
Prove that (1) (10) (4) (3) (2) (1). That is to say, (1), (2), (3), (4) and (10) are equivalent.
Prove that (1) (11) (7) (1). That is to say, (1), (7) and (11) are equivalent.
Prove that (7) (8) (9), (4) (5) (6), and (1) (12).