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Matrix Algebra THE INVERSE OF A MATRIX © 2012 Pearson Education, Inc.

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MATRIX OPERATIONS 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. In fact, C is uniquely determined by A, because if B were another inverse of A, then . This unique inverse is denoted by , so that and © 2012 Pearson Education, Inc.

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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 © 2012 Pearson Education, Inc.

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MATRIX OPERATIONS Theorem 5: If A is an invertible matrix, then for each b in , the equation has the unique solution Proof: Take any b in A solution exists because if is substituted for x, then So is a solution. To prove that the solution is unique, show that if u is any solution, then u must be If , we can multiply both sides by and obtain , , and © 2012 Pearson Education, Inc.

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MATRIX OPERATIONS Theorem 6:

If A is an invertible matrix, then is invertible and If A and B are 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, If A is an invertible matrix, then so is AT, and the inverse of AT is the transpose of That is, © 2012 Pearson Education, Inc.

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MATRIX OPERATIONS Proof: To verify statement (a), find a matrix C such that and These equations are satisfied with A in place of C. Hence is invertible, and A is its inverse. Next, to prove statement (b), compute: A similar calculation shows that For statement (c), use Theorem 3(d), read from right to left, Similarly, © 2012 Pearson Education, Inc.

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ELEMENTARY MATRICES Hence AT is invertible, and its inverse is .

The generalization of Theorem 6(b) is as follows: The product of invertible matrices is invertible, and the inverse is the product of their inverses in the reverse order. 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. © 2012 Pearson Education, Inc.

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ELEMENTARY MATRICES Example 1: Let , , ,

Compute E1A, E2A, and E3A, and describe how these products can be obtained by elementary row operations on A. © 2012 Pearson Education, Inc.

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ELEMENTARY MATRICES Solution: Verify that , , .

, , . Addition of times row 1 of A to row 3 produces E1A. © 2012 Pearson Education, Inc.

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ELEMENTARY MATRICES An interchange of rows 1 and 2 of A produces E2A, and multiplication of row 3 of A by 5 produces E3A. Left-multiplication by E1 in Example 1 has the same effect on any matrix. Since , we see that E1 itself is produced by this same row operation on the identity. © 2012 Pearson Education, Inc.

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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 Im. Each elementary matrix E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I. © 2012 Pearson Education, Inc.

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ELEMENTARY MATRICES Theorem 7: An matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into Proof: Suppose that A is invertible. Then, since the equation has a solution for each b (Theorem 5), 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 In. That is, © 2012 Pearson Education, Inc.

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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 E1, …, Ep such that . That is, (1) Since the product Ep…E1 of invertible matrices is invertible, (1) leads to © 2012 Pearson Education, Inc.

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ALGORITHM FOR FINDING Thus A is invertible, as it is the inverse of an invertible matrix (Theorem 6). Also, . Then , which says that results from applying E1, …, Ep successively to In. This is the same sequence in (1) that reduced A to In. Row reduce the augmented matrix If A is row equivalent to I, then is row equivalent to . Otherwise, A does not have an inverse. © 2012 Pearson Education, Inc.

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ALGORITHM FOR FINDING Example 2: Find the inverse of the matrix

, if it exists. Solution: © 2012 Pearson Education, Inc.

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ALGORITHM FOR FINDING © 2012 Pearson Education, Inc.

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ALGORITHM FOR FINDING Theorem 7 shows, since , that A is invertible, and . Now, check the final answer. © 2012 Pearson Education, Inc.

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ANOTHER VIEW OF MATRIX INVERSION

It is not necessary to check that since A is invertible. Denote the columns of In by e1,…,en. Then row reduction of to can be viewed as the simultaneous solution of the n systems , , …, (2) where the “augmented columns” of these systems have all been placed next to A to form . © 2012 Pearson Education, Inc.

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ANOTHER VIEW OF MATRIX INVERSION

The equation and the definition of matrix multiplication show that the columns of are precisely the solutions of the systems in (2). © 2012 Pearson Education, Inc.

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