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This module provides sample problems which develop concepts related to the identity and inverse matrices.

This assignment is brought to you by one of my favorite numbers, and I’m sure it’s one of yours…the number 1. Some people say that 1 is the loneliest number that you’ll ever do. (*Bonus: who said that?) But I say, 1 is the multiplicative identity.

Allow me to demonstrate.

You get the idea? 1 is called the multiplicative identity because it has this lovely property that whenever you multiply it by anything, you get that same thing back. But that’s not all! Observe…

The fun never ends! The point of all that was that every number has an inverse. The inverse is defined by the fact that, when you multiply a number by its inverse, you get 1.

Write the equation that defines two numbers a and b as inverses of each other.

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Find the inverse of 4 5 size 12{ { {4} over {5} } } {} .

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Find the inverse of –3.

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Is there any number that does not have an inverse, according to your definition in #7?

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So, what does all that have to do with matrices? (I hear you cry.) Well, we’ve already seen a matrix which acts as a multiplicative identity! Do these problems.

[ 3 8 -4 12 ] [ 1 0 0 1 ] =

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[ 1 0 0 1 ] [ 3 8 -4 12 ] =

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Pretty nifty, huh? When you multiply 1 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} ##0 {} # 1{} } right ]} {} by another 2×2 matrix, you get that other matrix back. That’s what makes this matrix (referred to as [ I ] ) the multiplicative identity.

Remember that matrix multiplication does not, in general, commute: that is, for any two matrices [ A ] and [ B ] , the product AB is not necessarily the same as the product BA. But in this case, it is: [ I ] times another matrix gives you that other matrix back no matter which order you do the multiplication in. This is a key part of the definition of I , which is…

Definition of [i]

The matrix I is defined as the multiplicative identity if it satisfies the equation: AI = IA = A

Which, of course, is just a fancy way of saying what I said before. If you multiply I by any matrix, in either order, you get that other matrix back.

We have just seen that 1 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} ##0 {} # 1{} } right ]} {} acts as the multiplicative identify for a 2×2 matrix.

  • A

    What is the multiplicative identity for a 3×3 matrix?
  • B

    Test this identity to make sure it works.
  • C

    What is the multiplicative identity for a 5×5 matrix? (I won’t make you test this one…)
  • D

    What is the multiplicative identity for a 2×3 matrix?
  • E

    Trick question! There isn’t one. You could write a matrix that satisfies AI = A , but it would not also satisfy IA = A —that is, it would not commute, which we said was a requirement. Don’t take my word for it, try it! The point is that only square matrices (*same number of rows as columns) have an identity matrix.
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So what about those inverses? Well, remember that two numbers a and b are inverses if a b = 1 . As you might guess, we’re going to define two matrices A and B as inverses if A B = [ I ] . Let’s try a few.

Multiply: 2 2 1 2 1 1 1 2 size 12{ left [ matrix { 2 {} # 2 { {1} over {2} } {} ##- 1 {} # - 1 { {1} over {2} } {} } right ]} {} 3 5 2 4 size 12{ left [ matrix { 3 {} # 5 {} ##- 2 {} # - 4{} } right ]} {}

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Multiply: 3 5 2 4 size 12{ left [ matrix { 3 {} # 5 {} ##- 2 {} # - 4{} } right ]} {} 2 2 1 2 1 1 1 2 size 12{ left [ matrix { 2 {} # 2 { {1} over {2} } {} ##- 1 {} # - 1 { {1} over {2} } {} } right ]} {}

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You see? These two matrices are inverses : no matter which order you multiply them in, you get [ I ] . We will designate the inverse of a matrix as A -1 which looks like an exponent but isn’t really, it just means inverse matrix—just as we used f -1 to designate an inverse function. Which leads us to…

Definition of a-1

The matrix A -1 is defined as the multiplicative inverse of A if it satisfies the equation: A -1 A = A A -1 = I (*where I is the identity matrix)

Of course, only a square matrix can have an inverse, since only a square matrix can have an I ! Now we know what an inverse matrix does , but how do you find one?

Find the inverse of the matrix 3 2 5 4 size 12{ left [ matrix { 3 {} # 2 {} ##5 {} # 4{} } right ]} {}

  • A

    Since we don’t know the inverse yet, we will designate it as a bunch of unknowns: a b c d size 12{ left [ matrix { a {} # b {} ##c {} # d{} } right ]} {} will be our inverse matrix. Write down the equation that defines this unknown matrix as our inverse matrix.
  • B

    Now, in your equation, you had a matrix multiplication. Go ahead and do that multiplication, and write a new equation which just sets two matrices equal to each other.
  • C

    Now, remember that when we set two matrices equal to each other, every cell must be equal. So, when we set two different 2x2 matrices equal, we actually end up with four different equations. Write these four equations.
  • D

    Solve for a , b , c , and d .
  • E

    So, write the inverse matrix A -1 .
  • F

    Test this inverse matrix to make sure it works!
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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Advanced algebra ii: activities and homework. OpenStax CNX. Sep 15, 2009 Download for free at http://cnx.org/content/col10686/1.5
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