### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Applied Linear Algebra MATH 2210

UCONN

GPA 3.97

### View Full Document

## 19

## 0

## Popular in Course

## Popular in Mathematics (M)

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.

## Reviews for Applied Linear Algebra

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I made $350 in just two days after posting my first study guide."

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.