Matrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.

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1 2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX

2 MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a iverse of A. I fact, C is uiquely determied by A, because if B were aother iverse of A, the B= BI = B( AC) = ( BA) C = IC = C This uique iverse is deoted by 1 A A = I AA 1 = I ad. A 1, so that. Slide 2.2-2

3 MATRIX OPERATIONS Theorem 5: If A is a ivertible matrix, the for each b i, the equatio A x= b has the uique 1 solutio. Proof: Take ay b i. A solutio exists because if is substituted for x, 1 1 the. So x = A b is a solutio. A 1 b Ax = A( A b) = ( AA )b= Ib= b A 1 b To prove that the solutio is uique, show that if u is ay solutio, the u must be. A u= b A Au If, we ca multiply both sides by ad obtai,, ad. = A b Iu = A 1 b 1 A A b u = A b Slide 2.2-3

4 MATRIX OPERATIONS Theorem 6: a. If A is a ivertible matrix, the is ivertible ad b. If A ad B are ivertible matrices, the so is AB, ad the iverse of AB is the product of the iverses of A ad B i the reverse order. That is, c. If A is a ivertible matrix, the so is A T, ad the iverse of A T is the traspose of A 1. That is, T 1 1 T ( A ) ( AB) 1 1 = = A B A ( A ) = ( A ) A 1 Slide 2.2-4

5 MATRIX OPERATIONS Proof: To verify statemet (a), fid a matrix C such that 1 A C = I ad CA 1 = These equatios are satisfied with A i place of C. Hece is ivertible, ad A is its iverse. A 1 Next, to prove statemet (b), compute: ( )( ) ( ) A similar calculatio shows that. For statemet (c), use Theorem 3(d), read from right 1 T T 1 T T to left,. Similarly,. Slide I AB B A = A BB A = AIA = AA = I 1 1 ( A ) A = ( AA ) = I = I T 1 T T A ( A ) = I = I ( B A )( AB) = I

6 ELEMENTARY MATRICES Hece A T is ivertible, ad its iverse is. The geeralizatio of Theorem 6(b) is as follows: 1 ( A ) The product of ivertible matrices is ivertible, ad the iverse is the product of their iverses i the reverse order. A ivertible matrix A is row equivalet to a idetity matrix, ad we ca fid A 1 by watchig the row reductio of A to I. A elemetary matrix is oe that is obtaied by performig a sigle elemetary row operatio o a idetity matrix. T Slide 2.2-6

7 ELEMENTARY MATRICES Example 1: Let E = 0 1 0,, 1 E = E = , a b c A= d e f g h i Compute E 1 A, E 2 A, ad E 3 A, ad describe how these products ca be obtaied by elemetary row operatios o A. Slide 2.2-7

8 ELEMENTARY MATRICES Solutio: Verify that a b c EA= d e f 1 g 4a h 4b i 4c a b c EA= d e f 3 5g 5h 5i. d e f EA= a b c 2 g h i,, 4 Additio of times row 1 of A to row 3 produces E 1 A. Slide 2.2-8

9 ELEMENTARY MATRICES A iterchage of rows 1 ad 2 of A produces E 2 A, ad multiplicatio of row 3 of A by 5 produces E 3 A. Left-multiplicatio by E 1 i Example 1 has the same effect o ay matrix. 3 E I = E Sice, we see that E 1 itself is produced by 1 1 this same row operatio o the idetity. Slide 2.2-9

10 ELEMENTARY MATRICES Example 1 illustrates the followig geeral fact about elemetary matrices. If a elemetary row operatio is performed o a m matrix A, the resultig matrix ca be writte as EA, where the m mmatrix E is created by performig the same row operatio o I m. Each elemetary matrix E is ivertible. The iverse of E is the elemetary matrix of the same type that trasforms E back ito I. Slide

11 ELEMENTARY MATRICES Theorem 7: A matrix A is ivertible if ad oly if A is row equivalet to I, ad i this case, ay sequece of elemetary row operatios that reduces A to I also trasforms I ito. A 1 Proof: Suppose that A is ivertible. A x= The, sice the equatio has a solutio for each b (Theorem 5), A has a pivot positio i every row. Because A is square, the pivot positios must be o the diagoal, which implies that the reduced echelo form of A is I. That is,. A I b Slide

12 ELEMENTARY MATRICES A I Now suppose, coversely, that. The, sice each step of the row reductio of A correspods to left-multiplicatio by a elemetary matrix, there exist elemetary matrices E 1,, E p such that A E 1 A E 2 (E 1 A)... E p (E p 1...E 1 A) = I E... E A= I That is, ----(1) p Sice the product E p E 1 of ivertible matrices is ivertible, (1) leads to 1. (E p...e 1 ) 1 (E p...e 1 )A = (E p...e 1 ) 1 I A = (E p...e 1 ) 1. Slide

13 ALGORITHM FOR FINDING A 1 Thus A is ivertible, as it is the iverse of a ivertible matrix (Theorem 6). Also, = (... ) =... p 1 p 1 A E E E E A 1 = E p...e 1 I The, which says that results from applyig E 1,..., E p successively to I. This is the same sequece i (1) that reduced A to I. [ ] Row reduce the augmeted matrix A I. If A is row equivalet to I, the A I is row equivalet to I A 1! "# $ %&. Otherwise, A does ot have a iverse.. A 1 Slide

14 ALGORITHM FOR FINDING Example 2: Fid the iverse of the matrix A = A 1, if it exists. Solutio:! "# A I! $ %& = # # # " $! & # & # & # % " $ & & & % Slide

15 ALGORITHM FOR FINDING " $ $ $ # " $ $ $ # " $ $ $ # / 2 2 1/ 2 A 1 % " ' $ ' $ ' $ & # % ' ' ' & / / / 2 2 1/ % ' ' ' & Slide % ' ' ' &

16 ALGORITHM FOR FINDING Theorem 7 shows, sice ad A I 9/2 7 3/2 = /2 2 1/2 1 A Now, check the fial aswer. A 1 1 AA, that A is ivertible, /2 7 3/ = = /2 2 1/ Slide

17 ANOTHER VIEW OF MATRIX INVERSION It is ot ecessary to check that ivertible. 1 A A = I sice A is Deote the colums of I by e 1,,e. The row reductio of A I to ca be viewed as the simultaeous solutio of the systems! "# [ ] A x= e 1 A x= e A x= e 2 I A 1,,, ----(2) where the augmeted colums of these systems have all bee placed ext to A to form A e 1 e 2 e $ %& =! A I $ "# %&. Slide

18 ANOTHER VIEW OF MATRIX INVERSION AA 1 = I The equatio ad the defiitio of matrix multiplicatio show that the colums of A 1 are precisely the solutios of the systems i (2). Slide

19 2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES

20 THE INVERTIBLE MATRIX THEOREM Theorem 8: Let A be a square matrix. The the followig statemets are equivalet. That is, for a give A, the statemets are either all true or all false. a. A is a ivertible matrix. b. A is row equivalet to the idetity matrix. c. A has pivot positios. d. The equatio has oly the trivial solutio. e. The colums of A form a liearly idepedet set. A x= 0 Slide 2.3-2

21 THE INVERTIBLE MATRIX THEOREM A x= f. The equatio has at least oe solutio for each b i. g. The colums of A spa. h. A T is a ivertible matrix. b Slide 2.3-3

22 THE INVERTIBLE MATRIX THEOREM Theorem 8 could also be writte as The equatio A x= b has a uique solutio for each b i. This statemet implies (b) ad hece implies that A is ivertible. The followig fact follows from Theorem 8. AB = B= A 1 Let A ad B be square matrices. If, the A ad B are both ivertible, with ad. The Ivertible Matrix Theorem divides the set of all matrices ito two disjoit classes: the ivertible (osigular) matrices, ad the oivertible (sigular) matrices. I A= B 1 Slide 2.3-4

23 THE INVERTIBLE MATRIX THEOREM Each statemet i the theorem describes a property of every ivertible matrix. The egatio of a statemet i the theorem describes a property of every sigular matrix. For istace, a sigular matrix is ot row equivalet to I, does ot have pivot positio, ad has liearly depedet colums. Slide 2.3-5

24 THE INVERTIBLE MATRIX THEOREM Example 1: Use the Ivertible Matrix Theorem to decide if A is ivertible: Solutio: " $ A $ $ # A = % " ' $ ' $ ' $ & # % ' ' ' & Slide 2.3-6

25 THE INVERTIBLE MATRIX THEOREM So A has three pivot positios ad hece is ivertible, by the Ivertible Matrix Theorem, statemet (c). The Ivertible Matrix Theorem applies oly to square matrices. For example, if the colums of a 4 3 matrix are liearly idepedet, we caot use the Ivertible Matrix Theorem to coclude aythig about the existece or oexistece of solutios of equatio of the form. A x= b Slide 2.3-7

26 INVERTIBLE LINEAR TRANSFORMATIONS Matrix multiplicatio correspods to compositio of liear trasformatios. Whe a matrix A is ivertible, the equatio ca be viewed as a statemet about liear trasformatios. See the followig figure. 1 A Ax= x Slide 2.3-8