Finding the Inverse of a Matrix. Properties of Matrices We have discovered that the commutative...

Finding the Inverse of a Matrix

Properties of Matrices

We have discovered that the commutative We have discovered that the commutative property for multiplication does not work for property for multiplication does not work for matrix multiplication. Let’s consider some matrix multiplication. Let’s consider some of the other properties of real numbers. Is of the other properties of real numbers. Is there a multiplicative identity for matrices? there a multiplicative identity for matrices? Is there a multiplicative inverse for Is there a multiplicative inverse for matrices?matrices?

The Multiplicative Identity

The multiplicative identity for real numbers The multiplicative identity for real numbers is the number 1. The property is:is the number 1. The property is:

In terms of matrices we need a matrix In terms of matrices we need a matrix that can be multiplied by a matrix (A) and that can be multiplied by a matrix (A) and give a product which is the same matrix give a product which is the same matrix (A).(A).

If a is a real number, then a x 1 = 1 x a = a.If a is a real number, then a x 1 = 1 x a = a.


10

012I

This matrix exists and it is called the This matrix exists and it is called the identity matrix. It is named identity matrix. It is named II and it comes and it comes in different sizes. It is a square matrix with in different sizes. It is a square matrix with all 1’s on the main diagonal and all other all 1’s on the main diagonal and all other entries are 0.entries are 0.

100

010

001

3I


04

52A

10

01

04

52Multiply Multiply AAII

aa1111= (-2)(1) + (5)(0) = -2 = (-2)(1) + (5)(0) = -2

aa1212= (-2)(0) + (5)(1) = 5 = (-2)(0) + (5)(1) = 5

aa2121= (4)(1) + (0)(0) = 4 = (4)(1) + (0)(0) = 4

aa2222= (4)(0) + (0)(1) = 0= (4)(0) + (0)(1) = 0

04

52

The Identity Matrix for Multiplication

Let Let AA be a square matrix with n rows be a square matrix with n rows and n columns. Let and n columns. Let II be a matrix with be a matrix with the same dimensions and with 1’s on the same dimensions and with 1’s on the main diagonal and 0’s elsewhere. the main diagonal and 0’s elsewhere.

Then Then AAII = = IIA = AA = A


1406

7410

2973

9470

B

Give the multiplicative identity for matrix B.Give the multiplicative identity for matrix B.

1000

0100

0010

0001

I

This identity matrix is This identity matrix is II4.4.

The Multiplicative Inverse

For every nonzero real number a, there is a For every nonzero real number a, there is a real number 1/a such that a(1/a) = 1.real number 1/a such that a(1/a) = 1.

In terms of matrices, the product of a In terms of matrices, the product of a square matrix and its inverse is square matrix and its inverse is II..

10

01

)3(1)1(2)2(1)1(2

)3(1)1(3)2(1)1(3

32

11

12

13

The Inverse of a Matrix

Let Let AA be a square matrix with be a square matrix with nn rows rows and and nn columns. If there is an columns. If there is an nn x x nn matrix matrix BB such that such that AB = AB = II and and BA = BA = II, , then then AA and and BB are inverses of one are inverses of one another. The inverse of matrix another. The inverse of matrix AA is is denoted by denoted by AA-1-1..


To show that matrices are inverses of one To show that matrices are inverses of one another, show that the multiplication of the another, show that the multiplication of the matrices is commutative and results in the matrices is commutative and results in the identity matrix.identity matrix.

Show that A and B are inverses.Show that A and B are inverses.

23

35

53

32BandA


10

01

)2(5)3(3)3(5)5(3

)2(3)3(2)3(3)5(2

23

35

53

32AB

and and


10

01

)5(2)3(3)3(2)2(3

)5)(3()3(5)3)(3()2(5

53

32

23

35BA

Finding the Inverse of a Matrix - Method 1

dc

baBandALet

53

21Use the equation AB = Use the equation AB = I.I.

Write and solve the equation:Write and solve the equation:

10

01

53

21

dc

ba

Inverses – Method 1, cont.

10

01

53

21

dc

ba

10

01

5353

22

dbca

dbca

1235

153

02

053

12

dandbcanda

db

db

ca

ca

Inverses – Method 1, cont.

13

25So the inverse of A = So the inverse of A =

We can check this by multiplying A x AWe can check this by multiplying A x A-1-1

10

01

)1(5)2(3)3(5)5(3

)1(2)2(1)3(2)5(1

13

25

53

21

Finding the Inverse with a Calculator

To find the inverse of a matrix using the To find the inverse of a matrix using the calculator, enter the matrix into the calculator, enter the matrix into the calculator and use the xcalculator and use the x-1-1 key. key.


012

431

112

B

36

48C

Find the inverse of each matrix using the Find the inverse of each matrix using the calculator.calculator.


This error message This error message means that the matrix means that the matrix does not have an does not have an inverse.inverse.

A matrix that does not have an inverse is A matrix that does not have an inverse is called an called an invertibleinvertible matrix. matrix.

DeterminantsDeterminants

Each square matrix can be Each square matrix can be assigned a real number called assigned a real number called the the determinantdeterminant of the matrix. of the matrix. It is denoted by the symbol It is denoted by the symbol ..

dc

ba

dc

baAIf

means the means the determinant determinant of A.of A.

DeterminantsDeterminants

The determinant of a 2 x 2 The determinant of a 2 x 2 matrix is found as follows:matrix is found as follows:

cbaddc

ba

Determinants

76

87GFind the determinant Find the determinant

of the matrix.of the matrix.

14849)8(6)7(776

87

Determinants

22

11H

Find the determinant of the matrix.Find the determinant of the matrix.

0)1(2)2(122

11

If the determinant If the determinant of a matrix = 0, the of a matrix = 0, the matrix does not matrix does not have an inverse. have an inverse. Matrix H is Matrix H is invertible.invertible.

Determinants can be used to find the inverse of a matrix.

ac

bd

AdetA

thenAdetanddc

baAIf

)(

1

,0)(

1

Determinants can be used to find the inverse of a matrix.

ac

bdis called the adjoint of the original matrix. Notice it is matrix. Notice it is

found by switching the entries on the main diagonal and changing the signs of the entries on the other diagonal.

Find the multiplicative Find the multiplicative inverse of:inverse of:

43

21A

21

23

12

13

24

2

11A

2)2(3)4(143

21

We can check to see if we are correct We can check to see if we are correct by multiplying. Remember that by multiplying. Remember that AAAA-1-1 = = II

10

01

)2/1(4)1(3)2/3(4)2(3

)2/1(2)1(1)2/3(2)2(1

21

23

12

43

21

Find the inverse using determinants.

11

31

21

21

23

21

30

12

31061

21

Find the inverseFind the inverse

No No inverseinverse

Recall that when the Recall that when the determinant of a matrix is 0 determinant of a matrix is 0 the matrix will not have an the matrix will not have an inverse because division by 0 inverse because division by 0 is undefined.is undefined.

42

84

Finding the determinant of a 3 x 3 matrix

Finding the determinant of a 3x3 matrix.

hg

edc

ig

fdb

ih

fea

ihg

fed

cba

One way to find the determinant of a 3x3 One way to find the determinant of a 3x3 matrix is the formula below.matrix is the formula below.

Find the determinant using the formula

420

513

502


2

3028

)6(5)12(0)14(2

)1(0)2(35)5(0)4(30)5(2)4(12

20

135

40

530

42

512

420

513

502


214

321

112


13

91014

)9(1)10(1)7(2

)2(4)1(11)3(4)2(11)3(1)2(22

14

211

24

31)1(

21

322

214

321

112

Finding the Inverse of a Matrix. Properties of Matrices We have discovered that the commutative...

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