topic 1 matrix

8/8/2019 Topic 1 Matrix

1/59

1

Topic 1

Matrix


2/59

2

451

687

253

What Is A Matrix?

A matrix is a rectangular collection

of like objects, usually numbers.W

eare primarily interested in matricesbecause they can be used to solvesystems of linear equations (covered

in week 3 & 4).

A =


3/59

3

What Is A Matrix? (cont.)

Other examples:


4/59

4

What Is A Matrix? (cont.)

Many notations to represent matrices:

In this class, we will use this notation:


5/59

5

451

687

5

The order AKA size of a matrix is thenumber of rows and columns.

the order of this example is 3 X 3

Read 3 by 3

Order Of A Matrix

row

column

row column


6/59

6

Square Matrix

A matrix is called a square matrix ifit has the same numbers of rows ascolumns.

3 X 3

it is a square matrix

4 X 3

it is NOT a square matrix

451

687

253

121

973

965

258


7/59

7

Column and Row Matrix

Column Matrix

A matrix th

ath

as only one column.

Row MatrixA matrix that has only one row.

3

2

8

726


8/59

8

Zero Matrix

A zero matrix is a matrix where all theelements are zeros.

000

000

000


9/59

9

Identity Matrix

The identity matrix is a matrix that hasone's on the diagonal, and zeros

everywhere else.

The identity matrix is usually written as"I".

100

010

001


10/59

10

Diagonal Of A Matrix

The diagonal of a matrix are theelements that have identical row andcolumn numbers

e.g. matrix 2 X 2, 3 X 3, etc.

A diagonal matrix is one that hasnon-zero elements only on thediagonal.

7

6

8


11/59

11**

**

***

******

**

**

*

000000

000000

00000

00000

00000

000000

000000

0000000

Block Diagonal Matrix

A block diagonal matrix is like a diagonalmatrix, except that elements exist in

the positions arranged as blocks.

(Where,

means a non-zero element.)


12/59

12

Band Matrix

A Band Matrix has numbers near thediagonal of the matrix, and nowhereelse. The width of the band is calledthe band width of the matrix.

***

****

*****

*****

*****

*****

****

***

00000

0000

000

000

000

000

0000

00000


13/59

13

Transpose Of A Matrix

The transpose of a matrix isobtained by interchanging the rowsand columns of matrix.

Example:

-

!

-

!

316

532

35

13

62

TA

A

-

!

-

!

205

614

123

261

012

543

TB

B


14/59

14

Orthogonal Matrix

If a matrix M has the property thatMTM = I than the matrix is called anOrthogonal matrix.


15/59

15

Symmetric Matrix

A symmetric matrix is a squarematrix equal to it's transpose, A = AT.

=B

=B

T

205

614

123

261

012

543

B is not symmetric matrix

653

542

321

653

542

321

-

-

=A

-

-

=A

A is a symmetric matrix


16/59

16

Triangular Matrix

A matrix with entries only below thediagonal, or with entries only abovethe diagonal, is called a (lower, upper)triangular matrix.

If the diagonal in those cases

consists only of 1's, then the matrixis unit triangular.


17/59

17

1

01

001

0001

00001

****

***

**

*

10000

1000

100

10

1

*

**

***

****

*****

****

***

**

*

0

00

000

0000

*

**

***

****

*****

0000

000

00

0

Triangular Matrix (cont.)

Upper Triangular Lower Triangular

Upper Unit Triangular Lower Unit Triangular

* Representsany numbers


18/59

18

Mathematical Operations

involving a Matrix and a ScalarWe can apply the following operationswhere one operand is a scalar (number) and

one is a matrix:Multiplication (x)

Division (/)


19/59

19

Mathematical Operations involving a

Matrix and aS

calar: Multiplication

Just multiply the scalar times eachelement in the matrix. This operationis commutative (the arrangement ofmatrixes can be switched)


20/59

20

Mathematical Operations involving a

Matrix and aS

calar : DivisionDivision only has meaning when amatrix is divided by a scalar.

Division of a scalar by a matrix is notdefined.


21/59

21

Mathematical Operations on Two

MatricesOnly three binary mathematicaloperations are defined:Addition

Subtraction

Multiplication


22/59

22


Matrices : AdditionAddition is only defined for twomatrices with the same order/size.Add the corresponding elements.Matrix addition is commutative.


23/59

23


Matrices : SubtractionSubtraction is just addition withunary inversion of the second matrix.

Subtraction is not commutative - theorder matters.Must be in the same order/size.


24/59

24


Matrices :MultiplicationMatrix multiplication is only definedwhen the second matrix has the samenumber of rows as the first matrixhas columns.

The resulting matrix has the same

number of rows as the first matrixand the same number of columns asthe second matrix.


25/59

25


Matrices :Multiplication (cont.)Here are some examples with matrices ofvarious orders.

Loosely speaking, multiplication is defined

when t

he middle numbers matc

h.


26/59

26

Basic Rules for Matrix

ArithmeticAddition, Subtraction, Multiplication & Division:

a and b are scalars and A, B, and C are matrices.


27/59


28/59

28

Example : Multiplication

Given A = and B = ,

Compute AB

-

320

124

-

5

3

1


29/59

29

Example : Multiplication

AB = =

= =

-

320

124

-

5

3

1

-

)5)(3()3)(2()1(0

)5)(1()3)(2()1(4

-

1560

564

-

21

3


30/59

30

Exercise 1

For the following matrices perform the indicatedoperation, if possible

1. A + B

2. B A

3. A + C

2 -4 -10 411 11 17 4

-2 -4 -4 013 -5 -3 14

Cant be done. Because of different sizes


31/59

31

Exercise 2

Compute

Given the matrices

15 55/2-7 -22/30 4


32/59

32

Exercise 3

Compute AC and CA for the following two matrices,if possible.

13 -53 17-56 -23 81

AC=

CA= cant be done, because of

the sizes


33/59

33

Exercise 4

Determine the transpose of these matrices:


34/59

34

The Identity Matrix

The identity matrix is a squarematrix with ones along the maindiagonal and zeros everywhere else.Here are the first few identitymatrices:


35/59

35

The Identity Matrix

The identity matrix can be multiplied by anysquare matrix and it leaves that matrix unchanged.

Multiplication by the identity matrix is alwayscommutative.

Here is an example.


36/59

36

Exercise 5

What is the answer?

Both will produceitself


37/59

37

The Determinant

The determinant is a scalar valueassigned to a square matrix.Therefore, non square matrices donot have a determinant.

The determinant of a (1x1) matrix is

just it's value, e.g. |5| = 5.


38/59

38

The Determinant of 2 X 2

The determinants of (2x2)

Example: Find the determinants for thegiven matrix.

Solution:

bcd,dc

ba

A!!

Adif

A24

12

-

-=

0

44

4122(det

!

!

!

-

)() -)(--A


39/59

39


To find determinant of 3 X 3, copy the first andsecond columns of matrix to form fourth and fifth

columns.

the formula to calculate the determinant is

aei + bfg + cdh gec hf a - idb

hg

ed

ba

ihg

fed

cba


40/59

40


Example: Find the determinants for thegiven matrix.

Solution: 013211

102

-=A

13

11

02

013

211

102

--

0

043100

010221113111320012det

=

-) --(-++=

))(()-)(()-)((-) -)(() +)(() +)((-A=


41/59

41

Exercise 6

Find the determinant of these matrices

A = 33

B = -467

C = 0


42/59

42

The Determinant & Singular

MatrixHigher order determinants are calculatedrecursively using the determinants of

smaller submatrices (discuss in Topic 3).A matrix with whose determinant has valuezero is called a singular matrix. If thedeterminant is not zero, the matrix is non-

singular.Exercise: Determine which matrices issingular and non-singular in the previousexercise A, B = non-singular

C = singular


43/59

43

The Matrix Inverse

DEFINITION:If A is a n x n matrix and an inverse of A is an n x nmatrix A-1, such that

AA-1= A-1A = I where I is the identity matrix.

then we call A invertible (non-singular) and we saythat A-1 is an inverse of the matrix A.

If we cant find such a matrix A-1 we call A asingular matrix (det A = zero).


44/59

44

The Matrix Inverse of 2 X 2

An inverse of 2 X 2 matrix:

dc

baA =if

ac-

b-d

bc-ad

1

=dc

ba

=A

1-

1-

then


45/59

45


Example: Find the inverse of the given matrix (ifit exists)

Solution:

Find its determinant = ad - bc =1(7) (2)(3)

= 1

A

73

21=

13

27

13

27

1

1

-

-=

-

-=

1-


46/59

46

Exercise 7

1. Find the determinant of this matrix.Determine if the following matrix is singular,based from its determinant.

2

. Find th

e determinant of th

is matrixand find its inverse. .

determinant= 0,therefore issingular

determinant= -10,inverse = -1/2 -1/5

1/2 2/5


47/59

47


If

Then steps to find A-1 are:1. Find determinant of 3X32. Find minor3. Find Cofactor4. Find Adjoint5. Replace results in formula

ihg

fed

cba

=A

adj(A)

A

1=A

1-


48/59

48

If then

Where

The Matrix Inverse of 3 X 3 :

Find Minor

ihg

fed

cba

=A

minorminorminor

minorminorminor

minorminorminor

minor

ihg

ed

cba

=A

fh-ei

ihg

fed

cba

aminor == fg-diihgfed

cba

bminor ==

eg-dh

ihg

fed

cba

cminor == ch-bi

ihg

fed

cba

dminor ==


49/59

49


Find Minor

cgai

ihg

fed

cba

emin == bgah

ihg

fed

cba

fmin ==

cebf

ihg

fed

cba

gmin == cdaf

ihg

fed

cba

hmin ==

bdae

ihg

fed

cba

imin ==


50/59

50


FindCofactor

add ve value to the circled elementin the minor matrix

minorminorminor

minorminorminor

minorminorminor

minor

ihg

fed

cba

=A

minorminorminor

minorminorminor

minorminorminor

i(h-

(f-e(d-

c(b-a

Cofactor

)

))

)

=


51/59

51


Find AdjointTranspose the cofactor to obtain theadjoint matrix

minominomino

minominomino

minominomino

i(h-g

(f-e(d-

c(b-a

Cofacto

)

)))

=

minorminorminor

minorminorminor

minorminorminor

i(f-c

(h-(b-

(d-a

Adjoint

)

))

)

=


52/59


53/59

53


ExampleSolution: Step 2: Find the Minor

326

101

011

-

-

-

=

minorminorminor

minorminorminor

minorminorminor

minor

ihgfed

cba

=

2202130

326

101

011

)=

-(-

)=

)()-(-

(=

-

-

-

=

mia

3636131

326

101

011

bmi ) =- () =)() -(-(=

-

-

-

=


54/59

54


Example2026021

326

101

011

=-) =)() -((=

-

-

-

= -cmi

3032031

326

101

011

= --) = -)() -((=

-

-

-

= -dmi

3036031

326

101

011

=-) =)() -((=

-

-

-

= -emin

4626121

326

101

011

= --) =)(-) -(-(=

-

-

-

=minf


55/59

55


Example10100

326

101

011

=-) =)() -((=

-

-

-

= 1-1-gmi

101101

326

101

011

= --) = -)() -((=

-

-

-

= 1-mi

10101

326

101

011

=(-) =)() -((=

-

-

-

= -1)1-imin

11-1

4-33-

23-2

ihg

fed

cba

min

min

min

min min min

min min min

min ==A


56/59

56


ExampleSolution: Step 3: Find the Cofactor

minorminorminor

minorminorminor

minorminorminor

i(h-

(f-e(d-

c(b-a

Cofactor

)

))

)

=

111

433

232

111433

232

=

)- -)- -)- -

)- -

=C ofactor


57/59

57


ExampleSolution: Step 4: Find the Adjoint

minominomino

minominomino

minominomino

i(f-c

(h-e(b-

g(d-a

djoint

)

))

)

=

142

133

132

111

433

232

Adjoint

T

==


58/59

58


ExampleSolution: Step 5: Replace results in formula

adj(A)A

1=A

1-

142

133

132

142

133

132

1

1

---

---

---

=

-

=1-


59/59

Exercise 8

Compute the inverse of the following matrix:

15/154 -5/154 -6/773/22 -1/22 1/1126/77 17/77 10/77

topic 1 matrix

Documents