Applied Linear Algebra
Applied Linear Algebra MATH 2210
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This 6 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 2210 at University of Connecticut taught by Thomas Roby in Fall. Since its upload, it has received 19 views. For similar materials see /class/205825/math-2210-university-of-connecticut in Mathematics (M) at University of Connecticut.
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Date Created: 09/17/15
22 The Inverse of a Matrix The inverse ofa real number a is denoted by a l For example 7 1 17 and 77 17171 An n x n matrixA is said to be invertible if there is an n x n matrix C satisfying CA AC In where In is the n x n identity matrix We call C the inverse ofA FACT lfA is invertible then the inverse is unique Proof Assume B and C are both inverses ofA Then BBB 1 C So the inverse is unique since any two inverses coincide The inverse ofA is usually denoted by A l We have 71147114 1quot Not all n x n matrices are invertible A matrix which is not invertible is sometimes called a singular matrix An invertible matrix is called nonsingular matrix Theorem 4 LetA a Z If adi bc 0 thenA is invertible and C d 7b A71 adibc C a If adi bc 0 thenA is not invertible Assume A is any invertible matrix and we wish to solve Ax b Then Ax b and so x orx Suppose w is also a solution to Ax b Then Aw b and Aw 7b which means w A 1 b 80 w A lb which is in fact the same solution We have proved the following result Theorem 5 lfA is an invertible n x n matrix then for each b in Rquot the equation Ax b has the unique solution X A lb 3xz2 72x2139 Solution Matrix form ofthe linear system 7 Z x1 I x2 EXAMPLE Use the inverse ofA 77 3 to solve 77x1 5 72 5x1 Theorem 6 Suppose A and B are invertible Then the following results hold a A 1 is invertible and A 1 1 A Le A is the inverse ofA l b AB is invertible and AB 1 B IA 1 c AT is invertible and Alf1 A 1T Partial proof of pan b ABXB IA I A A 1 A A71 Similarly one can show that B lA IXAB 1 Theorem 6 part b can be generalized to three or more invertible matrices ABCYI Earlier we saw a formula for nding the inverse of a 2 x 2 invertible matrix How do we find the inverse of an invertible n x n matrix To answer this question we first look at elementary matrices Elementary Matrices Definition An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix 100 100 100 EXAMPLE LetEl 020 E2 001 E3 010 and 001 010 301 abc A def ghz39 E1 E2 and E3 are elementary matrices Why Observe the following products and describe how these products can be obtained by elementary row operations on A E114 1 0 0 0 2 0 0 0 1 Q d g Q 139 abc 2d222f ghz39 abc ghz39 def a b c d e f 3ag 3bh 3ci If an elementary row operation is performed on an m x n matrix A the resulting matrix can be written as EA Where the m x m matrix E is created by performing the same row operations on 1quot Elementary matrices are invertible because row operations are reversible To determine the inverse of an elementary matrix E determine the elementary row operation needed to transform E back into 1 and apply this operation to 1 to nd the inverse For example WOl I Ol O l OO Eg1 1 X m 3 395 539 39 SP I ll l l 39 l o le l I O O O NIH O H l 3 D 3 100 100 100 EA 020 70 7301 001 010 010 100 100 0 E2E1A 001 7301 010 010 010 7301 100 00 100 E3E2E1A 010 10 010 301 7301 001 E3E2E1A 3 Then multiplying on the right by A l we get E3E2E1A 3 So 3E2E113 A71 The elementary row operations that row reduce A to In are the same elementary row operations that transform n into A l Theorem 7 An n x n matrixA is invertible if and only ifA is row equivalent to In and in this case any sequence of elementary row operations that reducesA to n will also transform n to A l Algorithm for finding A 1 Place A and 1 sidebyside to form an augmented matrix A I Then perform row operations on this matrix which will produce identical operations onA and 1 So by Theorem 7 A I will row reduce to I A l orA is not invertible 2 0 0 EXAMPLE FindtheinverseofA 73 01 ifitexists 0 10 Solution 2 0 010 o 10 0 o 0 A1 730 o10010001 0 10 0 01 0 01 0 8014 Ixle O NIH l i Order of multiplication is important EXAMPLE Suppose ABC and D are invertible n x n matrices and A BD 7quotC Solve forD in terms ofABC and D Solution 7A7 731 JACi D 7quot B IAC 1 D 7quot B lAC 1 D