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These notes closely follow the presentation of the material given in David C. Lay’s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working through the exercises therein. The Inverse of a Square Matrix Each n n identity matrix In 1 0 0 0 1 0 0 0 1 plays a role in matrix algebra similar to the role played by the number 1 in the regular algebra of numbers. In particular, if C is any m n matrix, then CI n C, and if D is any n m matrix, then I n D D. In the regular algebra of numbers, every real number a 0 has a unique multiplicative inverse. This means that there is a unique real number, a 1 such that aa 1 a 1 a 1. For example the multiplicative inverse of 5 is 1/5 (which we also denote by 5 1 ) because 5 5 1 5 1 5 1. We will ask this same type of questions for square matrices: Given an n n matrix, A, can we find an n n matrix B such that AB BA I n ? We begin by giving some definitions that apply to matrices that are not necessarily square. Definition If A is an m n matrix and C is a n m matrix such that CA I n , then C is said to be a left inverse of A. Definition If A is an m n matrix and D is a n m matrix such that AD I m , then D is said to be a right inverse of A. It was proved in homework problems 23–25 in Section 2.1 that if a matrix A has both a left and a right inverse, then A must be a square matrix and the left and right inverses of A must be equal to each other. In other words: If A has size m n, C and D both have size n m, CA I n , and AD I m , then m n and C D. Considering what has been said in the above paragraph, it only makes sense to talk about a matrix having (or not having) both a left and a right inverse if the matrix is square. However, it is possible for a non–square matrix to have a left inverse but no right inverse (or vice-versa). 1 Definition A matrix that has both a left and a right inverse is said to be an invertible matrix. Example The matrix 1 2 A 3 4 is invertible because the matrix B 2 1 3 2 12 is both a left and a right inverse of A. 2 Theorem If a matrix, A, is invertible, then A has a unique left inverse and a unique right inverse and these left and right inverses are equal to each other. (We call this unique matrix the inverse of A and denote it by A 1 .) Proof Suppose that B is a left inverse of A. Since A is invertible, we know that A also has a right inverse. However, we also know that every right inverse of A must be equal to B. In other words, B can be the only right inverse of A. But this means that every left inverse of A must equal B. Thus B is the only left inverse of A. We have proved that A has a unique left inverse. By similar reasoning, we can prove that A has a unique right inverse. It is also clear (from the reasoning in homework problems 23-25 of Section 2.1) that these left and right inverses must be equal to each other. Example Fort the matrix A 1 2 3 4 , the matrix B 2 1 3 2 12 is the only left inverse of A, and B is also the only right inverse of A. The matrix B is called the inverse of the matrix A and we can write A 1 B. 3 Theorem 1. If A is an invertible matrix, then A 1 is also an invertible matrix, and 1 A 1 A. 2. If A and B are invertible matrices of the same size, then AB is also an invertible matrix and AB 1 B 1 A 1 . 3. If A is an invertible matrix, then A T is an invertible matrix and T 1 A T A 1 . 4 Elementary Matrices An elementary matrix is a matrix that can be obtained from an identity matrix by performing a single elementary row operation. Example The matrices E 1 , E 2 , and E 3 shown below are all elementary matrices. 1 0 0 E1 0 0 1 0 1 0 1 2 0 1 0 0 , E2 0 3 0 0 0 1 , E3 0 1 0 . 0 0 1 Since every elementary row operation is reversible, all elementary matrices are invertible and their inverses are obtained by performing the reverse elementary row operation on the identity matrix. 5 Example To obtain the elementary matrix 1 0 0 E1 0 0 1 , 0 1 0 we interchange rows 1 and 2 of the identity matrix. Thus, to obtain E 1 1 , we interchange rows 1 and 2 of the identity matrix. Therefore, 1 0 0 E 1 1 0 0 1 . 0 1 0 To obtain the elementary matrix 1 0 0 E2 0 3 0 , 0 0 1 we scale row 2 of the identity matrix by a factor of 3. Thus, to obtain E 1 2 , we scale row 2 of the identity matrix by a factor of 1/3. Therefore, 1 0 0 E 1 2 0 1 3 0 . 0 0 1 To obtain the elementary matrix 1 2 0 E3 0 1 0 , 0 0 1 we replace row 1 of the identity matrix by (row 1 (-2 times row 2)). Thus, to obtain E 1 3 , we replace row 1 of the identity matrix by (row 1 (2 times row 2)).. Therefore, 1 2 0 E 1 3 0 1 0 . 0 0 1 6 Lemma Suppose that B is a matrix obtained by performing a single elementary row operation on the matrix A. Also, suppose that E is the elementary matrix obtained by performing this same elementary row operation on I. Then B EA. Example The matrix 1 3 6 B 11 4 1 1 2 2 is obtained by replacing row 2 of the matrix 1 3 6 A 9 0 3 1 2 2 with (row 2 plus (2 times row 3)). The elementary matrix 1 0 0 E 0 1 2 0 0 1 is obtained by replacing row 2 of the identity matrix with (row 2 plus (2 times row 3)). Observe that B EA. 7 Theorem An n n matrix, A, is invertible if and only if A~I n . In this case, any sequence of elementary row operations that transforms A into I n also transforms I n into A 1 . 8 An Algorithm for Finding A 1 To find the inverse of an invertible n n matrix A: 1. Form the matrix 2. Perform elementary row operations on A In transformed into I n . The result will be A In I n A 1 until A has been . Example Use the algorithm described above to find the inverse of the matrix A 1 1 1 0 0 1 0 1 1 2 4 0 1 0 . 0 0 9 The Inverse of a 2 2 Matrix Theorem If A a b , c d then A is invertible if and only if ad bc 0. If A is invertible, then A 1 d b 1 ad bc c a Example Let A 1 2 3 4 . Then ad bc 14 23 2 0 and A 1 1 2 4 2 3 1 2 1 3 2 12 . 10