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This module introduces the identity matrix and its properties.

When multiplying numbers, the number 1 has a special property: when you multiply 1 by any number, you get that same number back. We can express this property as an algebraic generalization:

1 x = x

The matrix that has this property is referred to as the identity matrix .

Definition of identity matrix

The identity matrix , designated as [ I ] , is defined by the property: [ A ] [ I ] = [ I ] [ A ] = [ A ]

Note that the definition of [I] stipulates that the multiplication must commute —that is, it must yield the same answer no matter which order you multiply in. This is important because, for most matrices, multiplication does not commute.

What matrix has this property? Your first guess might be a matrix full of 1s, but that doesn’t work:

1 2 3 4 size 12{ left [ matrix { 1 {} # 2 {} ##3 {} # 4{} } right ]} {} 1 1 1 1 size 12{ left [ matrix { 1 {} # 1 {} ##1 {} # 1{} } right ]} {} = 3 3 7 7 size 12{ left [ matrix { 3 {} # 3 {} ##7 {} # 7{} } right ]} {} so 1 1 1 1 size 12{ left [ matrix { 1 {} # 1 {} ##1 {} # 1{} } right ]} {} is not an identity matrix

The matrix that does work is a diagonal stretch of 1s, with all other elements being 0.

1 2 3 4 size 12{ left [ matrix { 1 {} # 2 {} ##3 {} # 4{} } right ]} {} 1 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} ##0 {} # 1{} } right ]} {} = 1 2 3 4 size 12{ left [ matrix { 1 {} # 2 {} ##3 {} # 4{} } right ]} {} so 1 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} ##0 {} # 1{} } right ]} {} is the identity for 2x2 matrices
2 5 9 π 2 8 3 1 / 2 8 . 3 size 12{ left [ matrix { 2 {} # 5 {} # 9 {} ##π {} # - 2 {} # 8 {} ## - 3 {} # 1/2 {} # 8 "." 3{}} right ]} {} 1 0 0 0 1 0 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} # 0 {} ##0 {} # 1 {} # 0 {} ## 0 {} # 0 {} # 1{}} right ]} {} = 2 5 9 π 2 8 3 1 / 2 8 . 3 size 12{ left [ matrix { 2 {} # 5 {} # 9 {} ##π {} # - 2 {} # 8 {} ## - 3 {} # 1/2 {} # 8 "." 3{}} right ]} {} 1 0 0 0 1 0 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} # 0 {} ##0 {} # 1 {} # 0 {} ## 0 {} # 0 {} # 1{}} right ]} {} is the identity for 3x3 matrices

You should confirm those multiplications for yourself, and also confirm that they work in reverse order (as the definition requires).

Hence, we are led from the definition to:

The identity matrix

For any square matrix, its identity matrix is a diagonal stretch of 1s going from the upper-left-hand corner to the lower-right, with all other elements being 0. Non-square matrices do not have an identity. That is, for a non-square matrix [ A ] , there is no matrix such that [ A ] [ I ] = [ I ] [ A ] = [ A ] .

Why no identity for a non-square matrix? Because of the requirement of commutativity. For a non-square matrix [ A ] you might be able to find a matrix [ I ] such that [ A ] [ I ] = [ A ] ; however, if you reverse the order, you will be left with an illegal multiplication.

Questions & Answers

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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