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Lecture 5 inverse of matrices - section 2-2 and 2-3
1.
2 2.2 © 2012 Pearson
Education, Inc. Math 337-102 Lecture 5 THE INVERSE OF A MATRIX
2.
Slide 2.2- 2©
2012 Pearson Education, Inc. Inverse Matrix - Definition An matrix A is said to be invertible if there is an matrix C such that and where , the identity matrix. In this case, C is an inverse of A. This unique inverse is denoted by A-1 A-1 A = I and AA-1 = I Invertible = nonsingular Not invertible = singular n n× n n× CA I= AC I= n I I= n n×
3.
Slide 2.2- 3©
2012 Pearson Education, Inc. Inverse of 2x2 Matrix Theorem 4: Let . If , then A is invertible and If , then A is not invertible. The quantity is called the determinant of A, and we write This theorem says that a matrix A is invertible iff detA ≠ 0. a b A c d = 0ad bc− ≠ 1 1 d b A c aad bc − − = −− 0ad bc− = ad bc− det A ad bc= − 2 2×
4.
Inverse of 2x2
Matrix - Example Find the inverse of A= Slide 2.2- 4© 2012 Pearson Education, Inc.
5.
Slide 2.2- 5©
2012 Pearson Education, Inc. Solving Equations with Inverse Matrices Theorem 5: If A is an invertible matrix, then for each b in Rn , the equation Ax = b has the unique solution x = A-1 b. Proof: n n×
6.
Solving Equations with
Inverse Matrices - Example Use inverse matrix to solve: 3x1 + 4x2 = 3 5x1 + 6x2= 7 Slide 2.2- 6© 2012 Pearson Education, Inc.
7.
Slide 2.2- 7©
2012 Pearson Education, Inc. Theorem 2-6 a) If A is an invertible matrix, then A-1 is invertible and (A-1 )-1 = A b) If A and B are nxn invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is, (AB)-1 = B-1 A-1 a) If A is an invertible matrix, then so is AT , and the inverse of AT is the transpose of A-1 . That is, (AT )-1 = (A-1 )T
8.
Slide 2.2- 8©
2012 Pearson Education, Inc. ELEMENTARY MATRICES An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix.
9.
Slide 2.2- 9©
2012 Pearson Education, Inc. Theorem 2-7 – Method for Finding A-1 An nxn matrix A is invertible iff A is row equivalent to In. The row operations that reduces A to In also transforms In into A-1 .
10.
Slide 2.2- 10©
2012 Pearson Education, Inc. ALGORITHM FOR FINDING Example 2: Find the inverse of the matrix , if it exists. Solution: A = 0 1 2 1 0 3 4 −3 8 1 A−
11.
Slide 2.2- 11©
2012 Pearson Education, Inc. ALGORITHM FOR FINDING . Now, check the final answer. 1 9 / 2 7 3/ 2 2 4 1 3/ 2 2 1/ 2 A− − − = − − − 1 0 1 2 9 / 2 7 3/ 2 1 0 0 1 0 3 2 4 1 0 1 0 4 3 8 3/ 2 2 1/ 2 0 0 1 AA− − − = − − = − − 1 A−
12.
Slide 2.3- 12©
2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM Theorem 8: Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false. a. A is an invertible matrix. b. A is row equivalent to the nxn identity matrix. c. A has n pivot positions. d.The equation Ax = 0 has only the trivial solution. e.The columns of A form a linearly independent set. f.The linear transformation x⟼Ax is one-to-one. g. Ax=b has at least one solution for each b in Rn .
13.
Slide 2.3- 13©
2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM (contd) h. The columns of A span Rn . i. The linear transformation x⟼Ax maps Rn onto Rn . j. There is an nxn matrix C such that CA=I. k. There is an nxn matrix D such that AD=I. l. AT is an invertible matrix.
14.
Slide 2.3- 14©
2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM The Invertible Matrix Theorem divides the set of all nxn matrices into two disjoint classes: the invertible (nonsingular) matrices noninvertible (singular) matrices. Each statement in the theorem describes a property of every nxn invertible matrix. The negation of a statement in the theorem describes a property of every nxn singular matrix. For instance, an nxn singular matrix is not row equivalent to In, does not have n pivot position, and has linearly dependent columns.
15.
Slide 2.3- 15©
2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM Example 1: Use the Invertible Matrix Theorem to decide if A is invertible: Solution: 1 0 2 3 1 2 5 1 9 A − = − − −
16.
Slide 2.3- 16©
2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM The Invertible Matrix Theorem applies only to square matrices. For example, if the columns of a 4x3 matrix are linearly independent, we cannot use the Invertible Matrix Theorem to conclude anything about the existence or nonexistence of solutions of equation of the form Ax=b.
17.
Slide 2.3- 17©
2012 Pearson Education, Inc. INVERTIBLE LINEAR TRANSFORMATIONS Matrix multiplication corresponds to composition of linear transformations. When a matrix A is invertible, the equation can be viewed as a statement about linear transformations. See the following figure. 1 x xA A− =
18.
Slide 2.3- 18©
2012 Pearson Education, Inc. INVERTIBLE LINEAR TRANSFORMATIONS A linear transformation T:Rn Rn is invertible if there exists a function S:Rn Rn such that S(T(x)) = x for all x in Rn T(S(x)) = x for all x in Rn Theorem 9: Let T:Rn Rn be a linear transformation with standard matrix A. Then T is invertible iff A is an invertible matrix. T-1 :x⟼A-1 x
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