calculus w2 t

7/29/2019 Calculus w2 t

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CALCULUS

week 2

created by Mira Musrini B

week2

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Pre Test

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1. Define the matrix .

2. if A and B are matrices with same values of its entriesbut have different sizes ,can sum of A+B be obtained?Explain!

3. if A and B are matrices with different values of its

entries but have same sizes ,can difference of A-B beobtained? Explain!

4. if A is 2x3 matrix and B is 3x2 matrix . is AB=BA ?Explain !


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Matrix

Definition:

Matrix is a rectangular array of numbers. The

numbers in the array are called the entries of matrix.

some example of matrices are :

week1Mira Musrini MT

3

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4,3

1,

000

02

13

2

,3012,

41

03

21

e


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Matrix


4 thus the general matrix 3x4 (A) might be written

as

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34333231

24232221

14131211

aaaa

aaaa

aaaa

3x4 , means size of row is 3 and size of column

is 4

A=


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matrix


5 Definition: Two matrices are defined to be equal if they have

he same size and their coresponding entries are equal.for example , consider the matrices

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043

012

53

12

3

12cB

xA

if x=5 ,then A=B ,but for all other values of x the matrix A

are not equal . there is no value of x for which A=C

since A and C have different sizes.


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operations on matrices

Definition: if A and B are matrices of the same size, then the sum A+Bis the matrix obtained by adding the entries of B to the coresponding

entries of A.

Matrices of diferent size cannot be added. Consider the matrices


6

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2211

5423

1022

1534

0724

4201

3012

CBA

53073221

4542

BA

the expressions A+C ,B+C are

undefined


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operations on matrices


7Definition: if A and B are matrices of the same size, then the difference

A-B is the matrix obtained by subtracting the entries of B from thecoresponding entries of A.

Matrices of diferent size cannot be subtracted. Consider the matrices

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22

11

54231022

1534

07244201

3012

CBA

51141

5223

2526

BA

the expressions A-C ,B-C

are undefined


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Scalar multiplications of matrix

Definitions: If A is any matrix and c is any scalar , then the

product cA is the matrix obtained by multiplying each entry of Aby c.

In matrix notation, if A=[aij ] , then (cA)ij=c(Aij)=caij .

Consider the matrices:


8

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401

123

3

1

531

720)1(

262

8642A

havewe,then

1203369

531720

131432

CB

CBA

It is common pratice to denote (-1)B by -B


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Multiplications of matrices

Definition: If A is an mxr matrix and B is rxn matrix then the

product AB is the mxn matrix whose entries are determined

as follows. To find the entry of row i and column j of AB, single

out row i of matrix A and column j from matrix B. Multiply

coresponding entries from the row and column together and

then add up the resulting products

Consider the matrices


9

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2572

13103414

062

421BA

Since A is 2x3 matrix and B is 3x4 matrix , the product AB is 2x3

matrix. To determine ,for example , the entry in row 2 and column3 of AB , we single out row 2 from A and column 3 from B.

The illustrations is given in the next slide .


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The computations of remaining

products are:

(1.4)+(2.0)+(4.2) = 12

(1.1)-(2.1)+(4.7) =27

(1.4)+(2.3)+(4.5) = 30

(2.4)+(6.0)+(0.2) =8

(2.1)-(6.1)+(0.7)=-4

(2.3)+(6.1)+(0.2) =12

122648

13302712

AB

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if A is general mxr matrix and B is general rxn matrix, the entry of

(AB)ij is given by :


12

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Properties of multiplications of matrices

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For real numbers we have ab=ba which is calledthe commutative law for multiplications. Formatrices ,however, AB is not equal to BA.

For example, AB is defined but BA is undefined.

This is the situations if A is 2x3 matrix and B is3x4 matrix then AB is 2x4 matrix, but on otherhand BA is not defined.

other cases , AB and BA are both defined but

have diferent sizes . This is the situations if A is2x3 matrix and B is 3x2 matrix, then AB is 2x2matrix and BA is 3x3 matrix.


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Properties of multiplications of matrices

consider:

BAABthus

03

63

411

21

givesgmultiplyin

03

21

32

01

BAAB

BA

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Properties of operations matrices


15

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Assuming that the sizes of matrices are such that the

indicated operations can be performed , the following rules ofmatrices arithmatic are valid.

(a)A+B=B+A (commutative law for addition)

(b) A+(B+C)=(A+B)+C (distributive law for addition)

(c) A(BC)=(AB)C (distributive law for multiplication)

(d) A(B+C)=AB+AC (left distributive law )

(e) (B+C)A=BA+CA (right distributive law)

(f) A(B-C) = AB-AC (right distributive law)

(g) (B-C)A = (BA-CA)

(h) a(B+C)=aB +aC

(i) a(B-C )= aB-aC


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Properties of operations matrices

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Assuming that the sizes of matrices are such that the

indicated operations can be performed , the following rules ofmatrices arithmatic are valid.

(j) (a+b)C=aC+bC

(k) (a-b)C = aC - bC

(l) a(bC) = (ab)C

(m) a(BC)=(aB)C = B(aC)


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Types of matrices

Zero matrices such as


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0000

0000

000

000

000

00

00


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Types of matrices


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indetity of matrices are square matrices with 1s on main

diagonal and os off the main diagonal .

1000

0100

0010

0001

100

010

001

10

01

A matrix of this form is called identity matrix and is denoted by I.

If it is important to emphasize size then we shall write In , for the

nxn matrix.

identity matrix can be identified as :


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19

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Types of matrices

jiifa

jiifaa

ij

ijij 1

0

Aaaa

aaa

aaa

aaaAI

Aaaa

aaa

aaa

aaaAI

aaa

aaaA

232221

131211

232221

131211

3

232221

131211

232221

131211

2

232221

131211

100

010

001and

10

01

then

matrixheconsider t


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Types of matrices

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Triangular matrix

A Square matrix in which all the entries above main diagonal are zero iscalled lower triangularmatrix. and a square matrix in which all the

entries below main diagonal are zero is called upper triangularmatrix.


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Types of matrices

Symetric matrix

A Square matrix A is called symmetric if A=AT

the following matrices are symmetric ,since each is equal to its own transpose:

4

3

2

1

000

000

000

000

705

034

541

53

37

d

d

d

d

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Types of matrices

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illustrations of symmetric matrix

f


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Types of matrices

inverse of matrices

Definition : if A is square matrix , and if a matrix B is the same size can be foundthat AB=BA=I then A is said to be invertible and B is called inverse of A

IBA

IAB

AB

10

01

31

52

21

53

and

10

01

21

53

31

52

cesin,31

52ofinverseanis

21

53

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Types of matrices

diagonal matrix

a Square matrix in which all of the

entries off the main diagonal are zero

is called the diagonal matrix:

8000

0000

0040

0006

100

010

001

50

02

nd

d

d

...00

::::

0...0

0...0

2

1

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A general nxn diagonal matrix D can be written as


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System of Linear Equations

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there are two types of System of linear Equations :

(a) non homogenous system of linear equations

(b) homogenous system of liniear equations

an example of non homogenous system of linear equations is :

an example of homogenous system of linear equations is :


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non homogenous system of Linear Equations

there are three posibilities of solutions of non homogenous Systemof linear equations :

(a)no solutions

(b)one solutions

(c)

infinitely many solutionsconsider the non homogenous system of liniear equations :

zero)bothnotand(

zero)bothnotand(

22222

11111

bacybxa

bacybxa

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the graph of these equations are lines ; call them l1 and l2 there are three

posilibilities

ill t ti h t f Li


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illustrations non homogenous system of Linear

Equations

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The lines l1 and l2 may be paralel, in which case there is no

intersections and consequently no solution to the system


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Equations

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The lines l1 and l2 may intersect at on ly one point, in which case the

system has only one solution


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Equations

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The lines l1 and l2 may conincide, in which case there are infinitely

many solutions to the system


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homogenous System of Linear Equations

there are two posibilities of solutions of homogenous System of linear

equations :

(a)one solutions (trivial )

(b)infinitely many solutions (non trivial)

consider the homogenous system of liniear equations

zero)bothnotare,(0

zero)bothnotare,(0

2222

1111

baybxa

baybxa

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the graph of the equations are lines trough the origin.


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illustrations homogenous System of Linear

Equations

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trivial solutions of homogenous system of liniear equations


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illustrations homogenous System of Linear

Equations

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nontrivial solutions of homogenous system of linear equations


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POST TEST

1.consider matrices

determine : 3A-5B+C

2. consider matrix

determine AB and BA

is AB equal to BA ?

102

101

109

121

201

413CBA

321

213

101

100

133

112

BA

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