The Invertible Matrix Theorem

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Math 240 TA: Shuyi Weng Winter 2017 February 6, 2017 The Invertible Matrix Theorem Theorem 1. Let A 2 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 2 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 2 Rn. n 5. If A is an n × n matrix such that the equation Ax = ei is consistent for each ei 2 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 2 Rn, then the columns of A are linearly independent. 11. If the equation Ax = b is consistent for every b 2 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 2 Rn such that Ax = b for some b 2 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.

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The Invertible Matrix Theorem
The Invertible Matrix Theorem Ryan C. Daileda Trinity University Linear Algebra Daileda The Invertible Matrix Theorem Introduction It is important to recognize when a square matrix is invertible. We can now characterize invertibility in terms of every one of the concepts we have now encountered. We will continue to develop criteria for invertibility, adding them to our list as we go. The invertibility of a matrix is also related to the invertibility of linear transformations, which we discuss below. Daileda The Invertible Matrix Theorem Theorem 1 (The Invertible Matrix Theorem) For a square (n × n) matrix A, TFAE: a. A is invertible. b. A has a pivot in each row/column. RREF c. A −−−→ I. d. The equation Ax = 0 only has the solution x = 0. e. The columns of A are linearly independent. f. Null A = {0}. g. A has a left inverse (BA = In for some B). h. The transformation x 7→ Ax is one to one. i. The equation Ax = b has a (unique) solution for any b. j. Col A = Rn. k. A has a right inverse (AC = In for some C). l. The transformation x 7→ Ax is onto. m. AT is invertible. Daileda The Invertible Matrix Theorem Inverse Transforms Definition A linear transformation T : Rn → Rn (also called an endomorphism of Rn) is called invertible iff it is both one-to-one and onto. If [T ] is the standard matrix for T , then we know T is given by x 7→ [T ]x. The Invertible Matrix Theorem tells us that this transformation is invertible iff [T ] is invertible.
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