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1 © 2012 Pearson Education, Inc. Matrix Algebra THE INVERSE OF A MATRIX
Slide 2.2- 2 © 2012 Pearson Education, Inc. ELEMENTARY MATRICES An invertible matrix A is row equivalent to an identity matrix, and we can find by watching the row reduction of A to I. An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix.
Slide 2.2- 3 © 2012 Pearson Education, Inc. ELEMENTARY MATRICES Example 1: Let,,, Compute E 1 A, E 2 A, and E 3 A, and describe how these products can be obtained by elementary row operations on A.
Slide 2.2- 4 © 2012 Pearson Education, Inc. ELEMENTARY MATRICES An interchange of rows 1 and 2 of A produces E 2 A, and multiplication of row 3 of A by 5 produces E 3 A. Left-multiplication by E 1 in Example 1 has the same effect on any matrix. Since, we see that E 1 itself is produced by this same row operation on the identity.
Slide 2.2- 5 © 2012 Pearson Education, Inc. ELEMENTARY MATRICES Example 1 illustrates the following general fact about elementary matrices. If an elementary row operation is performed on an matrix A, the resulting matrix can be written as EA, where the matrix E is created by performing the same row operation on I m. Each elementary matrix E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I.
Slide 2.2- 6 © 2012 Pearson Education, Inc. ELEMENTARY MATRICES Theorem 7: An matrix A is invertible if and only if A is row equivalent to I n, and in this case, any sequence of elementary row operations that reduces A to I n also transforms I n into. Proof: Suppose that A is invertible. Then, since the equation has a solution for each b, A has a pivot position in every row. Because A is square, the n pivot positions must be on the diagonal, which implies that the reduced echelon form of A is I n. That is,.
Slide 2.2- 7 © 2012 Pearson Education, Inc. ELEMENTARY MATRICES Now suppose, conversely, that. Then, since each step of the row reduction of A corresponds to left-multiplication by an elementary matrix, there exist elementary matrices E 1, …, E p such that. That is, ----(1) Since the product E p …E 1 of invertible matrices is invertible, (1) leads to.
Slide 2.2- 8 © 2012 Pearson Education, Inc. MATRIX OPERATIONS 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 if and only if det.
Slide 2.2- 9 © 2012 Pearson Education, Inc. ALGORITHM FOR FINDING Example 2: Find the inverse of the matrix, if it exists.
1 © 2012 Pearson Education, Inc. Matrix Algebra CHARACTERIZATIONS OF INVERTIBLE MATRICES
Slide 2.3- 11 © 2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM Theorem 8: 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. a. A is an invertible matrix. b. A is row equivalent to the identity matrix. c. A has n pivot positions. d.The equation has only the trivial solution. e.The columns of A form a linearly independent set.
Slide 2.3- 12 © 2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM f.The linear transformation is one-to- one. g.The equation has at least one solution for each b in. h.The columns of A span. i.The linear transformation maps onto. j.There is an matrix C such that. k.There is an matrix D such that. l. A T is an invertible matrix.
Slide 2.3- 13 © 2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM Theorem 8 could also be written as “ The equation has a unique solution for each b in. ” This statement implies (b) and hence implies that A is invertible. The following fact follows from Theorem 8. Let A and B be square matrices. If, then A and B are both invertible, with and. The Invertible Matrix Theorem divides the set of all matrices into two disjoint classes: the invertible (nonsingular) matrices, and the noninvertible (singular) matrices.
Slide 2.3- 14 © 2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM Each statement in the theorem describes a property of every invertible matrix. The negation of a statement in the theorem describes a property of every singular matrix. For instance, an singular matrix is not row equivalent to I n, does not have n pivot position, and has linearly dependent columns.
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:
Slide 2.3- 16 © 2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM So A has three pivot positions and hence is invertible, by the Invertible Matrix Theorem, statement (c). The Invertible Matrix Theorem applies only to square matrices. For example, if the columns of a 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.
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.
Slide 2.3- 18 © 2012 Pearson Education, Inc. INVERTIBLE LINEAR TRANSFORMATIONS A linear transformation is said to be invertible if there exists a function such that for all x in ----(1) for all x in ----(2) 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 equation (1) and (2).
Slide 2.3- 19 © 2012 Pearson Education, Inc. INVERTIBLE LINEAR TRANSFORMATIONS Proof: Suppose that T is invertible. The (2) shows that T is onto, for if b is in and, then, so each b is in the range of T. Thus A is invertible, by the Invertible Matrix Theorem, statement (i). Conversely, suppose that A is invertible, and let. Then, S is a linear transformation, and S satisfies (1) and (2). For instance,. Thus, T is invertible.
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