lec3 linear equations

8/3/2019 Lec3 Linear Equations

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Determinant of a Matrix:

The determinant of a matrix is a scalar and is denoted as |A| or det(A). Thedeterminant has very important mathematical properties, but it is very

difficult to provide a substantive definition. For covariance and correlationmatrices, the determinant is a number that is sometimes used to expressthe generalizedvariance of the matrix. That is, covariance matrices withsmall determinants denote variables that are redundant or highlycorrelated. Matrices with large determinants denote variables that areindependent of one another.

A determinant of second order is denoted by

14)2*3()5*4(52

34

det21122211

2221

1211

aaaaaa

aaAD


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Third Order Determinant

7206

620

26

423

20

461

206

462

031

det

2322

1312

31

3332

1312

21

3332

2322

11

333231

232221

131211

D

aa

aaa

aa

aaa

aa

aaa

aaa

aaa

aaa

AD


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Determinant of any order

jnjnjjjj

rcrr

c

c

CaCaCaD

aaa

aaa

aaa

AD

...

...

............

...

...

det

2211

21

22221

11211

(j = 1,2,..,or n)


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General Properties of Determinantsa. Interchange of two rows multiplies tha value of the determinant by -1.b. Addition of multiple of a row to another row does not alter the value of

the determinant.

c. Multiplication of a row by c multiplies the value of the determinant by c.

d. Transposition leaves the value of a determinant unaltered.

e. A zero row or column renders the value of a determinant zero.

f. Proportional rows or columns render the value of a determinant zero. Inparticular, a determinant with two identical rows or column has the value zero.


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Cramers Method:

A method for solving a linear system of

equations using determinants. Cramers rule mayonly be used when the system is square and thecoefficient matrix is invertible.

x = D x / Dy = D y / D
http://www.mathwords.com/l/linear_system_of_equations.htmhttp://www.mathwords.com/l/linear_system_of_equations.htmhttp://www.mathwords.com/d/determinant.htmhttp://www.mathwords.com/s/square_system_of_equations.htmhttp://www.mathwords.com/c/coefficient_matrix.htmhttp://www.mathwords.com/i/invertible_matrix.htmhttp://www.mathwords.com/i/invertible_matrix.htmhttp://www.mathwords.com/c/coefficient_matrix.htmhttp://www.mathwords.com/s/square_system_of_equations.htmhttp://www.mathwords.com/d/determinant.htmhttp://www.mathwords.com/l/linear_system_of_equations.htmhttp://www.mathwords.com/l/linear_system_of_equations.htm


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Solution to Systems of Linear EquationsA system of linear equation:

133

843

1524

321

321

321

xxx

xxx

xxx

Gaussian Elimination

A systematic method of eliminating the unknowns and solving the equationsby back-substitution

Makes use of the basic matrix operations such as row interchanges,elimination by addition or subtraction, scalar multiplication, etc.


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Solution to Systems of Linear Equations

rcrr

c

c

aaa

aaa

aaa

A

...

............

...

...

21

22221

11211

Transform the system of linear equations into its equivalent matrix form

A x = bA bwhereA an r x c matrix which is compose of the coefficients of the unknownsx vector of unknownsb vector of the sum of the products of the unknowns and its

corresponding coefficient


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Solution to Systems of Linear EquationsConsider the system of equations below

133

843

1524

321

321

321

xxx

xxxxxx

In Matrix notation


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Solution to Systems of Linear EquationsThe augmented matrix which is used in matrix operations is the combinationof the matrix of coefficients and vector of known.

138

15

34

1

1113

24


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Solution to Systems of Linear EquationsIn Gaussian elimination, the objective is to transform the matrix into an uppertriangular matrix by matrix operations

3

2

1

33

23

13

22

1211

00

0

b

b

b

a

a

a

a

aa


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Solution to Systems of Linear EquationsRow operation (subtraction)

To eliminate a21

122

1121

*

/

RfRR

aaf

133

1131

*

/

RfRR

aaf

To eliminate a31


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Solution to Systems of Linear EquationsThe last step in the Gaussian elimination is the back substitution.

3

2

1

33

23

13

22

1211

00

0

b

b

b

a

a

a

a

aa

3333

2323222

1312212111

00

0

bxa

bxaxa

bxaxaxa

11

2123131

1

22

32322

33

3

3

a

xaxabx

a

xabx

a

bx


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Solution to Systems of Linear EquationsGauss Elimination: The three Possible Cases of Systems

Infinitely many solutions exist

Ex. Solve the linear three equations in four unknowns:

1.24.23.03.02.1

7.24.55.15.16.0

0.80.50.20.20.3

4321

4321

4321

xxxx

xxxx

xxxx

1.2

7.2

0.8

4.2

4.5

0.5

3.0

5.1

0.2

3.02.1

5.16.0

0.20.3

1.1

1.1

0.8

4.4

4.4

0.5

1.1

1.1

0.2

1.10

1.10

0.20.3

Sol:

First Step. Elimination of x1from the second and thirdequations by adding

-0.6/3.0 times the first eq. to the second eq.-1.2/3.0 times the first eq. to the third eq.

This gives:


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Solution to Systems of Linear EquationsSecond Step. Elimination of x2 from the third equations by adding

1.1/1.1 times the second eq. to the third eq.

0

1.1

0.8

0

4.4

0.5

0

1.1

0.2

00

1.10

0.20.3


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443

63

22

321

321

321

xxx

xxx

xxx

4

6

2

4

1

2

31

13

11

2

12

2

2

7

2

20

20

11

10

12

2

5

7

2

00

20

11

Solve the linear system

Sol:

First Step. Elimination of x1 from the second and third equationsgives

Second Step. Elimination of x2 from the third equation gives

Beginning with the last equation, we obtainsuccessively x3 = 2, x2 = -1, x1 = 1.

Gauss Elimination if a unique solution exist


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Solution to Systems of Linear EquationsGauss Elimination if no solution exist

6426

02

323

321

321

321

xxx

xxx

xxx

6

0

3

4

1

1

26

12

23

0

2

3

2

3/1

1

20

3/10

23

Sol:First Step. Elimination of x1 from the second and third equations

gives

Second Step. Elimination of x2 from the third equation gives

12

2

3

0

3/1

1

00

3/10

23

This shows that the system has nosolution


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Solution to Systems of Linear EquationsGauss Elimination. Partial pivoting

Solve the system

26826:

8253:

728:

3213

3212

321

xxxExxxE

xxE

728

8253

26826

32

321

321

xxxxx

xxx

7

5

29

2

2

8

80

40

26

7

8

26

2

2

8

80

53

26

Sol:We must pivot since E1 has no x1 term. In column 1, eq E3 has the largest

coefficient. Hence, we interchange E1 and E3.

Eliminate of x1 from the other equation gives


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The largest coefficient in column 2 is 8. Hence we take the new third equationas the pivot eq., interchanging eqs. 2 and 3,


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7

26

3

2

8

00

80

26

5

7

26

2

2

8

40

80

26

4

1

2

1

1

2

3

x

x

x

Eliminate x2

Back substitution yields,


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Consider a linear system.

Solution:


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This is a triangular matrix. Its associated system is

Clearly we have v= 1. Set z=s and w=t, then wehave

The first equation implies


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Using algebraic manipulations, we get

Putting all the stuff together, we have


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2X1 3X2 + 4X3 = 3-X1 + 4X2 +3X3 = 11

4X1 + 6X2 5X3 = -8


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Gauss Jordan eliminates the backwardsubstitution..


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A matrix must be square to have an inverse, but not all square matrices

have an inverse. In some cases, the inverse does not exist. For covarianceand correlation matrices, an inverse will always exist, provided that thereare more subjects than there are variables and that every variable has avariance greater than 0.

Matrix Inverse:In scalar algebra, the inverse of a number is that number which, whenmultiplied by the original number, gives a product of 1. Hence, the inverse

of x is simple 1/x. or, inslightly different notation, x-1

. In matrix algebra,the inverse of a matrix is that matrixwhich, when multiplied by the originalmatrix, gives an identity matrix. The inverse of a matrix is denoted by thesuperscript -1. Hence, AA-1 = A-1A = I

rcrr

c

c

T

jk

aaa

aaa

aaa

AA

AA

...

........ ... .

...

...

det

1

det

1

21

22221

11211

1 where Ajk is the cofactorof ajk in det A.


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Matrix Inverse:

8

31

13

1341

13

7

43

11

10det

431

113

211

13

12

11

A

A

A

A

A

231

11

241

21

243

21

23

22

21

A

A

A

21311

713

21

311

21

23

32

31

A

A

A

2.02.08.0

7.02.03.1

3.02.07.0

1

A


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A civil engineer involved in construction requires 6000, 5000,and 8000 m3 of sand, gravel, and coarse gravel, respectively,for building project. There are three pits from which thesematerials can be obtained. The composition of these pits

How many cubic meters must be hauled from each pit inorder to meet the engineers needs? Use at least 2 from anymatrix method.

Sand% Fine Gravel% CoarseGravel%Pit 1 32 30 38Pit 2 25 40 35Pit 3 35 15 50

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