<< Chapter < Page Chapter >> Page >
In this section, you will:
  • Find the inverse of a matrix.
  • Solve a system of linear equations using an inverse matrix.

Nancy plans to invest $10,500 into two different bonds to spread out her risk. The first bond has an annual return of 10%, and the second bond has an annual return of 6%. In order to receive an 8.5% return from the two bonds, how much should Nancy invest in each bond? What is the best method to solve this problem?

There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix.

Finding the inverse of a matrix

We know that the multiplicative inverse of a real number a is a −1 , and a a −1 = a −1 a = ( 1 a ) a = 1. For example, 2 −1 = 1 2 and ( 1 2 ) 2 = 1. The multiplicative inverse of a matrix    is similar in concept, except that the product of matrix A and its inverse A −1 equals the identity matrix    . The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by I n where n represents the dimension of the matrix. [link] and [link] are the identity matrices for a 2 × 2 matrix and a 3 × 3 matrix, respectively.

I 2 = [ 1 0 0 1 ]
I 3 = [ 1 0 0 0 1 0 0 0 1 ]

The identity matrix acts as a 1 in matrix algebra. For example, A I = I A = A .

A matrix that has a multiplicative inverse has the properties

A A −1 = I A −1 A = I

A matrix that has a multiplicative inverse is called an invertible matrix . Only a square matrix may have a multiplicative inverse, as the reversibility, A A −1 = A −1 A = I , is a requirement. Not all square matrices have an inverse, but if A is invertible, then A −1 is unique. We will look at two methods for finding the inverse of a 2 × 2 matrix and a third method that can be used on both 2 × 2 and 3 × 3 matrices.

The identity matrix and multiplicative inverse

The identity matrix    , I n , is a square matrix containing ones down the main diagonal and zeros everywhere else.

I 2 = [ 1 0 0 1 ] I 3 = [ 1 0 0 0 1 0 0 0 1 ]          2 × 2                  3 × 3

If A is an n × n matrix and B is an n × n matrix such that A B = B A = I n , then B = A −1 , the multiplicative inverse of a matrix     A .

Showing that the identity matrix acts as a 1

Given matrix A , show that A I = I A = A .

A = [ 3 4 −2 5 ]

Use matrix multiplication to show that the product of A and the identity is equal to the product of the identity and A.

A I = [ 3 4 −2 5 ] [ 1 0 0 1 ] = [ 3 1 + 4 0 3 0 + 4 1 −2 1 + 5 0 −2 0 + 5 1 ] = [ 3 4 −2 5 ]
A I = [ 1 0 0 1 ] [ 3 4 −2 5 ] = [ 1 3 + 0 ( −2 ) 1 4 + 0 5 0 3 + 1 ( −2 ) 0 4 + 1 5 ] = [ 3 4 −2 5 ]
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Given two matrices, show that one is the multiplicative inverse of the other.

  1. Given matrix A of order n × n and matrix B of order n × n multiply A B .
  2. If A B = I , then find the product B A . If B A = I , then B = A −1 and A = B −1 .

Showing that matrix A Is the multiplicative inverse of matrix B

Show that the given matrices are multiplicative inverses of each other.

A = [ 1 5 −2 −9 ] , B = [ −9 −5 2 1 ]

Multiply A B and B A . If both products equal the identity, then the two matrices are inverses of each other.

A B = [ 1 5 −2 −9 ] · [ −9 −5 2 1 ] = [ 1 ( −9 ) + 5 ( 2 ) 1 ( −5 ) + 5 ( 1 ) −2 ( −9 ) −9 ( 2 ) −2 ( −5 ) −9 ( 1 ) ] = [ 1 0 0 1 ]
B A = [ −9 −5 2 1 ] · [ 1 5 −2 −9 ] = [ −9 ( 1 ) −5 ( −2 ) −9 ( 5 ) −5 ( −9 ) 2 ( 1 ) + 1 ( −2 ) 2 ( −5 ) + 1 ( −9 ) ] = [ 1 0 0 1 ]

A and B are inverses of each other.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask