2.6 The Inverse of a Square Matrix


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1 200/2/6 page CHAPTER 2 Matrices and Systems of Linear Equations i i 2i 5 A = A = i i 4 + i i + i + 5i 26 The Inverse of a Square Matrix In this section we investigate the situation when, for a given n n matrix A, there exists a matrix B satisfying AB = I n and BA = I n (26) and derive an efficient method for determining B (when it does exist) As a possible application of the existence of such a matrix B, consider the n n linear system Ax = b (262) Premultiplying both sides of (262) by an n n matrix B yields Assuming that BA = I n, this reduces to (BA)x = Bb x = Bb (26) Thus, we have determined a solution to the system (262) by a matrix multiplication Of course, this depends on the existence of a matrix B satisfying (26), and even if such a matrix B does exist, it will turn out that using (26) to solve n n systems is not very efficient computationally Therefore it is generally not used in practice to solve n n systems However, from a theoretical point of view, a formula such as (26) is very useful We begin the investigation by establishing that there can be at most one matrix B satisfying (26) for a given n n matrix A Theorem 26 Let A be an n n matrix Suppose B and C are both n n matrices satisfying AB = BA = I n, (264) AC = CA = I n, (265) respectively Then B = C Proof From (264), it follows that C = CI n = C(AB) That is, C = (CA)B = I n B = B, where we have used (265) to replace CA by I n in the second step Since the identity matrix I n plays the role of the number in the multiplication of matrices, the properties given in (26) are the analogs for matrices of the properties xx =, x x =, which holds for all (nonzero) numbers x It is therefore natural to denote the matrix B in (26) by A and to call it the inverse of A The following definition introduces the appropriate terminology
2 200/2/6 page 6 26 The Inverse of a Square Matrix 6 DEFINITION 262 Let A be an n n matrix If there exists an n n matrix A satisfying AA = A A = I n, then we call A the matrix inverse to A, orjustthe inverse of A We say that A is invertible if A exists Invertible matrices are sometimes called nonsingular, while matrices that are not invertible are sometimes called singular Remark It is important to realize that A denotes the matrix that satisfies AA = A A = I n It does not mean /A, which has no meaning whatsoever 2 0 Example 26 If A = 2, verify that B = is the inverse of A 0 Solution: By direct multiplication, we find that AB = 2 = 00 = I 0 00 and BA = 2 = 00 = I 0 00 Consequently, (26) is satisfied, hence B is indeed the inverse of A We therefore write 0 A = 0 We now return to the n n system of Equations (262) Theorem 264 If A exists, then the n n system of linear equations Ax = b has the unique solution for every b in R n x = A b
3 200/2/6 page CHAPTER 2 Matrices and Systems of Linear Equations Proof We can verify by direct substitution that x = A b is indeed a solution to the linear system The uniqueness of this solution is contained in the calculation leading from (262) to (26) Our next theorem establishes when A exists, and it also uncovers an efficient method for computing A Theorem 265 An n n matrix A is invertible if and only if rank(a) = n Proof If A exists, then by Theorem 264, any n n linear system Ax = b has a unique solution Hence, Theorem 259 implies that rank(a) = n Conversely, suppose rank(a) = n We must establish that there exists an n n matrix X satisfying AX = I n = XA Let e, e 2,, e n denote the column vectors of the identity matrix I n Since rank(a) = n, Theorem 259 implies that each of the linear systems Ax i = e i, i =, 2,,n (266) has a unique solution x i Consequently, if we let X =[x, x 2,,x n ], where x, x 2,,x n are the unique solutions of the systems in (266), then that is, A[x, x 2,,x n ]=[Ax,Ax 2,,Ax n ]=[e, e 2,,e n ]; We must also show that, for the same matrix X, AX = I n (26) XA = I n Postmultiplying both sides of (26) by A yields That is, (AX)A = A A(XA I n ) = 0 n (268) Now let y, y 2,,y n denote the column vectors of the n n matrix XA I n Equating corresponding column vectors on either side of (268) implies that Ay i = 0, i =, 2,,n (269) But, by assumption, rank(a) = n, and so each system in (269) has a unique solution that, since the systems are homogeneous, must be the trivial solution Consequently, each y i is the zero vector, and thus XA I n = 0 n Therefore, XA = I n (260) Notice that for an n n system Ax = b, if rank(a) = n, then rank(a # ) = n
4 200/2/6 page The Inverse of a Square Matrix 65 Equations (26) and (260) imply that X = A We now have the following converse to Theorem 264 Corollary 266 Let A be an n n matrix If Ax = b has a unique solution for some column nvector b, then A exists Proof If Ax = b has a unique solution, then from Theorem 259, rank(a) = n, and so from the previous theorem, A exists Remark In particular, the above corollary tells us that if the homogeneous linear system Ax = 0 has only the trivial solution x = 0, then A exists Other criteria for deciding whether or not an n n matrix A has an inverse will be developed in the next three chapters, but our goal at present is to develop a method for finding A, should it exist Assuming that rank(a) = n, letx, x 2,,x n denote the column vectors of A Then, from (266), these column vectors can be obtained by solving each of the n n systems Ax i = e i, i =, 2,,n As we now show, some computation can be saved if we employ the GaussJordan method in solving these systems We first illustrate the method when n = In this case, from (266), the column vectors of A are determined by solving the three linear systems Ax = e, Ax 2 = e 2, Ax = e The augmented matrices of these systems can be written as A 0, A 0, A 0 0, 0 0 respectively Furthermore, since rank(a) = by assumption, the reduced rowechelon form of A is I Consequently, using elementary row operations to reduce the augmented matrix of the first system to reduced rowechelon form will yield, schematically, A 0 ERO 0 which implies that the first column vector of A is a x = a 2 a Similarly, for the second system, the reduction A 0 ERO 0 00a 00a 2, 00a 00b 00b 2 00b
5 200/2/6 page CHAPTER 2 Matrices and Systems of Linear Equations implies that the second column vector of A is b x 2 = b 2 b Finally, for the third system, the reduction A 0 00c 0 ERO 00c 2 00c implies that the third column vector of A is x = Consequently, a b c A =[x, x 2, x ]= a 2 b 2 c 2 a b c The key point to notice is that in solving for x, x 2, x we use the same elementary row operations to reduce A to I We can therefore save a significant amount of work by combining the foregoing operations as follows: A 00 00a b c 00 ERO 00a 2 b 2 c a b c The generalization to the n n case is immediate We form the n 2n matrix [A I n ] and reduce A to I n using elementary row operations Schematically, c c 2 c [A I n ] ERO [I n A ] This method of finding A is called the GaussJordan technique Remark Notice that if we are given an n n matrix A, we likely will not know from the outset whether rank(a) = n, hence we will not know whether A exists However, if at any stage in the row reduction of [A I n ] we find that rank(a) < n, then it will follow from Theorem 265 that A is not invertible Example 26 Find A if A = Solution: Using the GaussJordan technique, we proceed as follows
6 200/2/6 page 6 26 The Inverse of a Square Matrix 6 Thus, A = We leave it as an exercise to confirm that AA = A A = I A ( ) 2 A 2 ( ), A 2 ( 2) M ( /4) 4 A ( ), A 2 ( 2) Example 268 Continuing the previous example, use A to solve the system x + x 2 + x = 2, x 2 + 2x =, x + 5x 2 x = 4 Solution: The system can be written as Ax = b, where A is the matrix in the previous example, and 2 b = 4 Since A is invertible, the system has a unique solution that can be written as x = A b Thus, from the previous example we have x = = 4 Consequently, x = 5, x 2 =, and x = 2, so that the solution to the system is ( 5, ), 2 We now return to more theoretical information pertaining to the inverse of a matrix Properties of the Inverse The inverse of an n n matrix satisfies the properties stated in the following theorem, which should be committed to memory: 5 2 Theorem 269 Let A and B be invertible n n matrices Then A is invertible and (A ) = A 2 AB is invertible and (AB) = B A A T is invertible and (A T ) = (A ) T
7 200/2/6 page CHAPTER 2 Matrices and Systems of Linear Equations Proof The proof of each result consists of verifying that the appropriate matrix products yield the identity matrix We must verify that A A = I n and AA = I n Both of these follow directly from Definition We must verify that (AB)(B A ) = I n and (B A )(AB) = I n We establish the first equality, leaving the second equation as an exercise We have We must verify that (AB)(B )(A ) = A(BB )A = AI n A = AA = I n A T (A ) T = I n and (A ) T A T = I n Again, we prove the first part, leaving the second part as an exercise First recall from Theorem 222 that A T B T = (BA) T Using this property with B = A yields A T (A ) T = (A A) T = I T n = I n The proof of property 2 of Theorem 269 can easily be extended to a statement about invertibility of a product of an arbitrary finite number of matrices More precisely, we have the following Corollary 260 Let A,A 2,,A k be invertible n n matrices Then A A 2 A k is invertible, and (A A 2 A k ) = A k A k A Proof The proof is left as an exercise (Problem 28) Some Further Theoretical Results Finally, in this section, we establish two results that will be required in Section 2 and also in a proof that arises in Section 2 Theorem 26 Let A and B be n n matrices If AB = I n, then both A and B are invertible and B = A Proof Let b be an arbitrary column nvector Then, since AB = I n,wehave A(Bb) = I n b = b Consequently, for every b, the system Ax = b has the solution x = Bb But this implies that rank(a) = n To see why, suppose that rank(a) < n, and let A denote a rowechelon form of A Note that the last row of A is zero Choose b to be any column
8 200/2/6 page The Inverse of a Square Matrix 69 nvector whose last component is nonzero Then, since rank(a) < n, it follows that the system A x = b is inconsistent But, applying to the augmented matrix [A b ] the inverse row operations that reduced A to rowechelon form yields [A b] for some b Since Ax = b has the same solution set as A x = b, it follows that Ax = b is inconsistent We therefore have a contradiction, and so it must be the case that rank(a) = n, and therefore that A is invertible by Theorem 265 We now establish that 8 A = B Since AB = I n by assumption, we have A = A I n = A (AB) = (A A)B = I n B = B, as required It now follows directly from property of Theorem 269 that B is invertible with inverse A Corollary 262 Let A and B be n n matrices If AB is invertible, then both A and B are invertible Proof If we let C = B(AB) and D = AB, then AC = AB(AB) = DD = I n It follows from Theorem 26 that A is invertible Similarly, if we let C = (AB) A, then CB = (AB) AB = I n Once more we can apply Theorem 26 to conclude that B is invertible Exercises for 26 Key Terms Inverse, Invertible, Singular, Nonsingular, GaussJordan technique Skills Be able to check directly whether or not two matrices A and B are inverses of each other Be able to find the inverse of an invertible matrix via the GaussJordan technique Be able to use the inverse of a coefficient matrix of a linear system in order to solve the system Know the basic properties related to how the inverse operation behaves with respect to itself, multiplication, and transpose (Theorem 269) TrueFalse Review For Questions 0, decide if the given statement is true or false, and give a brief justification for your answer If true, you can quote a relevant definition or theorem from the text If false, provide an example, illustration, or brief explanation of why the statement is false An invertible matrix is also known as a singular matrix 8 Note that it now makes sense to speak of A, whereas prior to proving in the preceding paragraph that A is invertible, it would not have been legal to use the notation A
9 200/2/6 page 0 0 CHAPTER 2 Matrices and Systems of Linear Equations 2 Every square matrix that does not contain a row of 5 zeros is invertible 9 A = 2 26 A linear system Ax = b with an n n invertible coefficient matrix A has a unique solution 00 4 If A is a matrix such that there exists a matrix B with 0 A = 00 AB = I n, then A is invertible 02 5 If A and B are invertible n n matrices, then so is 42 A + B A = If A and B are invertible n n matrices, then so is AB 2 If A is an invertible matrix such that A 2 = A, then A 2 A = 26 2 is the identity matrix 4 8 If A is an n n invertible matrix and B and C are n n i 2 matrices such that AB = AC, then B = C A = + i 2i 2 2i 5 9 If A isa5 5matrix of rank 4, then A is not invertible 0 If A isa6 6 matrix of rank 6, then A is invertible 2 4 A = 2 4 Problems For Problems verify by direct multiplication that the 2 given matrices are inverses of one another [ ] [ ] 5 A = A =,A = [ ] [ ] 49 2 A =,A = 4 6 A = A = 2,A = Let 2 4 A = 5 2 For Problems 4 6, determine A, if possible, using the GaussJordan method If A exists, check your answer by verifying that AA = I n [ ] 2 4 A = [ + i 5 A = i [ i 6 A = + i 2 [ ] 00 A = A = ] ] Find the second column vector of A without determining the whole inverse For Problems 8 22, use A to find the solution to the given system x + x 2 =, 2x + 5x 2 = x + x 2 2x = 2, x 2 + x =, 2x + 4x 2 x = x 2ix 2 = 2, (2 i)x + 4ix 2 = i
10 200/2/6 page 26 The Inverse of a Square Matrix 2 22 x + 4x 2 + 5x =, 2x + 0x 2 + x =, 4x + x 2 + 8x = x + x 2 + 2x = 2, x + 2x 2 x = 24, 2x x 2 + x = 6 An n n matrix A is called orthogonal if A T = A For Problems 2 26, show that the given matrices are orthogonal [ ] 0 2 A = 0 [ ] /2 /2 24 A = /2 /2 [ cos α sin α 25 A = sin α cos α 26 A = + 2x 2 ] 2x 2x 2 2x 2x 2 2x 2x 2 2x 2 Complete the proof of Theorem 269 by verifying the remaining properties in parts 2 and 28 Prove Corollary 260 For Problems 29 0, use properties of the inverse to prove the given statement 29 If A is an n n invertible symmetric matrix, then A is symmetric 0 If A is an n n invertible skewsymmetric matrix, then A is skewsymmetric Let A be an n n matrix with A 4 = 0 Prove that I n A is invertible with (I n A) = I n + A + A 2 + A 2 Prove that if A, B, C are n n matrices satisfying BA = I n and AC = I n, then B = C If A, B, C are n n matrices satisfying BA = I n and CA = I n, does it follow that B = C? Justify your answer 4 Consider the general 2 2 matrix [ ] a a A = 2 a 2 a 22 and let = a a 22 a 2 a 2 with a 0 Show that if 0, A = [ ] a22 a 2 a 2 a The quantity defined above is referred to as the determinant of A We will investigate determinants in more detail in the next chapter 5 Let A be an n n matrix, and suppose that we have to solve the p linear systems Ax i = b i, i =, 2,,p where the b i are given Devise an efficient method for solving these systems 6 Use your method from the previous problem to solve the three linear systems if Ax i = b i, i =, 2, A = 2 4, b =, 6 2 b 2 = 2, b = 5 2 Let A be an m n matrix with m n (a) If rank(a) = m, prove that there exists a matrix B satisfying AB = I m Such a matrix is called a right inverse of A (b) If A = [ ], 24 determine all right inverses of A For Problems 8 9, reduce the matrix [A I n ] to reduced rowechelon form and thereby determine, if possible, the inverse of A A = A is a randomly generated 4 4 matrix For Problems 40 42, use builtin functions of some form of technology to determine rank(a) and, if possible,a 5 40 A =
11 200/2/6 page 2 2 CHAPTER 2 Matrices and Systems of Linear Equations Consider the n n Hilbert matrix 4 A = [ ] H n =, i, j n i + j 42 A is a randomly generated 5 5 matrix 4 For the system in Problem 2, determine A and use it to solve the system (a) Determine H 4 and show that it is invertible (b) Find H 4 and use it to solve H 4 x = b if b = [2,,, 5] T 2 Elementary Matrices and the LU Factorization We now introduce some matrices that can be used to perform elementary row operations on a matrix Although they are of limited computational use, they do play a significant role in linear algebra and its applications DEFINITION 2 Any matrix obtained by performing a single elementary row operation on the identity matrix is called an elementary matrix In particular, an elementary matrix is always a square matrix In general we will denote elementary matrices by E If we are describing a specific elementary matrix, then in keeping with the notation introduced previously for elementary row operations, we will use the following notation for the three types of elementary matrices: Type : P ij permute rows i and j in I n Type 2: M i (k) multiply row i of I n by the nonzero scalar k Type : A ij (k) add k times row i of I n to row j of I n Example 22 Write all 2 2 elementary matrices Solution: From Definition 2 and using the notation introduced above, we have Permutation matrix: P 2 = 2 Scaling matrices: M (k) = Row combinations: A 2 (k) = [ ] 0 0 [ ] k 0, M 0 2 (k) = [ ] 0, A k 2 (k) = [ ] 0 0 k [ ] k 0 We leave it as an exercise to verify that the n n elementary matrices have the following structure:
2.5 Gaussian Elimination
page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3
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