MAT225 Section Summary: 2.3

Characterizations of Invertible Matrices

Summary

Theorem 8: The Invertible Matrix Theorem

Let A be a square 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 invertible.
2. A is row equivalent to the identity matrix.
3. A has n pivot positions.
4. The equation has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation is one-to-one.
7. The equation has at least one solution for each b in .
8. The columns of A span .
9. The linear transformation maps onto .
10. There is an matrix C such that CA=I.
11. There is an matrix D such that AD=I.
12. is invertible.

As the author says, ``the power of the Invertible Matrix Theorem lies in the connections it provides between so many important concepts....''

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A linear transformation is said to be invertible if there exists a function such that

 

Theorem 9: Let be a linear transformation and let A be the standard matrix for T. Then T is invertible if and only if A is an invertible matrix. In that case, the linear transformation S given by is the unique function satisfying (1).

#38, p. 133

Definition: A matrix that is nearly - but not quite - singular is said to be ill-conditioned. A matrix that is ill-conditioned causes trouble when the time comes to invert, and for other calculations. The condition number of a matrix measures how poorly conditioned a matrix is. The identity matrix has a condition number of 1, and is perfectly well-conditioned. The larger the condition number is, the closer a matrix is to singular (the condition number is infinite for a singular matrix). For a matrix, the closer the determinant is to zero, the larger the condition number.

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