Math 240 TA: Shuyi Weng Winter 2017 February 6, 2017

The Invertible Matrix Theorem

Theorem 1. Let A ∈ Rn×n. Then the following statements are equivalent.

1. A is invertible.

2. A is row equivalent to In. 3. A has n pivots in its reduced echelon form. 4. The matrix equation Ax = 0 has only the trivial solution. 5. The columns of A are linearly independent. 6. The linear transformation T defined by T (x) = Ax is one-to-one.

7. The equation Ax = b has at least one solution for every b ∈ Rn. 8. The columns of A span Rn. 9. The linear transformation T defined by T (x) = Ax is onto.

10. There exists an n × n matrix B such that AB = In.

11. There exists an n × n matrix C such that CA = In. 12. AT is invertible.

Theorem 2. Let T : Rn → Rn be defined by T (x) = Ax. Then T is one-to-one and onto if and only if A is an invertible matrix. Problem. True or false (all matrices are assumed to be n × n, unless otherwise specified).

1. The identity matrix is invertible. 2. If A can be row reduced to the identity matrix, then it is invertible. 3. If both A and B are invertible, so is AB.

4. If A is invertible, then the matrix equation Ax = b is consistent for every b ∈ Rn. n 5. If A is an n × n matrix such that the equation Ax = ei is consistent for each ei ∈ R a column of the n × n identity matrix, then A is invertible. 6. If both A and B are invertible, then the inverse of AB is A−1B−1. 7. Every elementary matrix is invertible. " # a b 8. If A = , and ab − cd = 0, then A is invertible. c d 9. If A is invertible, so is A−1.

10. If the equation Ax = b is consistent for every b ∈ Rn, then the columns of A are linearly independent.

11. If the equation Ax = b is consistent for every b ∈ Rn, then the columns of A span Rn. 12. If A is invertible, so is AT , and (AT )−1 = (A−1)T . 13. If Ax = 0 has more than one solution, then A is invertible.

14. If there exists x ∈ Rn such that Ax = b for some b ∈ Rn, then A is invertible.

15. If Ax1 = Ax2 only when x1 = x2, then A is invertible. Solution to the T/F problems:

1. True 4. True 7. True 10. True 13. False 2. True 5. True 8. False 11. True 14. False 3. True 6. False 9. True 12. True 15. True