The Inverse of a Square Matrix


 Millicent Jennings
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1 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 inclass presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working through the exercises therein Each n n identity matrix The Inverse of a Square Matrix I n 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 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 IfAisanm 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 IfAisanm 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 in Section 21 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 viceversa) 1
2 Definition A matrix that has both a left and a right inverse is said to be an invertible matrix Example The matrix is invertible because the matrix A B is both a left and a right inverse of A
3 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 of Section 21) that these left and right inverses must be equal to each other Example Fort the matrix A , the matrix B 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
4 Theorem 1 If A is an invertible matrix, then A 1 is also an invertible matrix, and A 1 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 A T 1 A 1 T 4
5 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 E 1, E , E 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
6 Example To obtain the elementary matrix E 1 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, E 1 1 To obtain the elementary matrix E we scale row 2 of the identity matrix by a factor of 3 Thus, to obtain E 2 1, we scale row 2 of the identity matrix by a factor of 1/3 Therefore, E 2 1 To obtain the elementary matrix E we replace row 1 of the identity matrix by (row 1 (2 times row 2)) Thus, to obtain E 3 1, we replace row 1 of the identity matrix by (row 1 (2 times row 2)) Therefore, E ,,, 6
7 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 B is obtained by replacing row 2 of the matrix A with (row 2 plus (2 times row 3)) The elementary matrix E is obtained by replacing row 2 of the identity matrix with (row 2 plus (2 times row 3)) Observe that B EA 7
8 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
9 An Algorithm for Finding A 1 To find the inverse of an invertible n n matrix A: 1 Form the matrix A I n 2 Perform elementary row operations on A I n until A has been transformed into I n The result will be I n A 1 Example Use the algorithm described above to find the inverse of the matrix A
10 The Inverse of a 2 2 Matrix Theorem If a b A, c d then A is invertible if and only if ad bc 0 If A is invertible, then A 1 1 ad bc d c b a Example Let A Then ad bc and A
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DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
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