2 2 TheInverseof a Matrix

Qo when does a lineartransformationhave an inverse

oranyfunction

recall let f A B be afunction Thenthefunction

ofB 7A is an inversefor f if

gof x X forany XcA

fog y y foranyycB

If g exoto then it isunique call it f

A f BA f B

7 oror or

or aso

f fofT n

f of A BD id

Iidf t

what couldgowrong

A f Bfwheredoes

as f sendthis

problem t is not injective

Af B c wheredoes

2 f sendthisa

problem f is notsurjective

theorem Afunctionhas an inverse ifandonlyif itis booth one to oneandanto

back to lineartransformations

fromHw if the lineartransformationT.IR 7112 is

one to one andarto then n m So weonlyneed

to worry about T.IR 112

what about the matrices

let Turn 3112 Silk 3112 be lineartransformations

with standard mutinies nm A and B If S is

theinverse of T five S T then

Sot idn cud To S idn

The standard matures then satisfytwo equations

BA In and AB In

i e B A and

A A In AA I

Definite An mn matrix A is inventive if

there is another run matrix A such that

A A In AA t

unmpresi

let A Sy This is invertible the

inverse is a i f3,4g note normally A

andA t aren't sosimilar

checkthis

Aa f It I v

A a I Il 51 1If A

f o Tg then A a not invertible

since it's not square

let A log Then A is not invertible

this is dearShuethecolumns ofA are linearly

dependent but we can alsocheekdirectly

If therewere an inverse A

for some a 6 c d EIR ther

IE AA Y EI

r ra

This is notpossible so A is not invertible

consequences ofminertibility

Let A be an min invertiblematrix

The lineartransformation T 112 7112definedbyTLE AI hur an inverse

a lineartransformations w standardmatrixASo T is one to one andarto

For any bER thematrixequation

A8 5 has a uniquesolution

if AI E then A AI A E

A A I A E

In E A f

otherproperties

A 5 A

AB B A 1

cheek wehave

AB AB Inmultibothby A on left

A AB AB A In1 wiIn A 1

In B AB A t

B

B AB A

Mutt beB on left

B B AB I B A

AB B A I

AT A IT

Q How can wetellwhen A is invertibleif anechelonform ofA bus a prot inevery row column then A is invertibleHow can wefindA

morecorphrated

start with simple case 2 2

theorem let A f ba Then A is invertible

if andonly if ad bc 0 in which case

A fexercise checkthat AA Iz A A

example let A Ig Then ad be film 4 I D14 to

so A is invertible Weknow then that

a I it I

notes a act be is the determinant of A

warning muchmore complicated

forlargermetrics

why do we need ad bc fomotore

re skit L la Iif ad be o ther d f f c ba I

if c andd notbothzero then

thecolumns of A arelinearlydependent

so AD not invertible

if c d 0 then A fao

sothecolumns of A donotspun1122are so A is notinventive

recall A is invertible thereducedechelonform ofA is In

idea thereis a sequence of me opentrans Wtnhsends A run In if we represent rowoperatorsas some matrox E such that EA isthe resultofdoingthoseoperations to A then EA e In

andtherefore E A

elementary row operations as maturer

mc ir H Iswap 1stand2nd rows

scaling

r ir H iscale 1st row by 6

replacement

I viii It ii Ireplace 3rdrow with 312 1123

Def m An elementary matrix is an nxn matrix

obtainedfun In by applying an elementaryrow operation

NOTE elementary mutnut une invertible

So if Air invertible thenthereis a sequence

of elementary row operation whrh reduce A to InSome each is representedby a matron there is a

sequence of elementary manner E Ek sumthat9 r

frit operates lastoperation

Ew ErE A In

Some En Ez E and A areboth men maturer

thor means that A e Ele EZE

idea Ew EzE is the resultof applying to row

operations In the sameorder as A ur In to In

SE the row operation reducing A MO Incan applied toget In mo A

we'lldo both simultaneously

ALGORITHM FOR FINDINGA 1

flow reduce the augmentedmatrix A1 In

to get In 1A

example let A f Find A

Wehave

rain I lr

I po 2 I o l 3

I po O I z a 3

I o o Z I 4

I Iiif I I A

A e f Icheck this