term paper(mth 103),re3901b50

8/7/2019 Term Paper(MTH 103),RE3901B50

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MTH 103: BASIC MATHEMATICS - IMTH 103: BASIC MATHEMATICS - I

TERM PAPERTERM PAPER

Submitted by: -

Name: - Aparnesh Roy

Course: - BCA MCA

Section: - E3901

Roll No.: - RE3901B50

Group: - 2

Registration No.: - 10904928

Submitted to: - Lect. Syamsundhar Raju


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SOLUTION OF LINEAR PAIRSOLUTION OF LINEAR PAIR

OF EQUATIONS BYOF EQUATIONS BY

CRAMERS RULE ANDCRAMERS RULE ANDMATRIX METHODMATRIX METHOD


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ACKNOWLEDGEMENTACKNOWLEDGEMENT

I am very thankful to Lect.I am very thankful to Lect.

Syamsundhar sir, who gave meSyamsundhar sir, who gave me

advice on making term paperadvice on making term paperon Solution of Linear Pair ofon Solution of Linear Pair of

Equations by Cramers RuleEquations by Cramers Rule

And Matrix Method and alsoAnd Matrix Method and also

thanks to all my friends whothanks to all my friends who

helped me collectinghelped me collecting

information regarding my terminformation regarding my term

paper. This term paper ispaper. This term paper is

totally based on Finding outtotally based on Finding out

solutions of linear equationssolutions of linear equations

through Cramers Rule andthrough Cramers Rule and


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Matrix method and helped meMatrix method and helped me

in understanding this topic.in understanding this topic.

-: CONTENTS :--: CONTENTS :-

1) Introduction(a) Linear Equation

(b) Linear Equation in Two

Variables

2) Solution of Linear Equation

3) Methods for Finding out

Solutions

(1) Cramers Rule

(2) Matrix Method


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INTRODUCTIONINTRODUCTION

Linear Equation: -Linear Equation: -A linear equation is

an algebraic equation in which each term iseither a constant or the product of aconstant and (the first power of) a singlevariable.

Linear equations can have one or morevariables. Linear equations occur with great

regularity in applied mathematics. Whilethey arise quite naturally when modellingmany phenomena, they are particularlyuseful since many non-linear equations maybe reduced to linear equations by assumingthat quantities of interest vary to only asmall extent from some "background" state.

Linear Equation in Two Variables:Linear Equation in Two Variables:--

A common form of a linear equation in thetwo variables x and y is

y = mx + b,


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where m and b designate constants. Theorigin of the name "linear" comes from thefact that the set of solutions of such an

equation forms a straight line in the plane. Inthis particular equation, the constant mdetermines the slope or gradient of that lineand the constant term b determines the

point at which the line crosses the y-axis,otherwise known as the y-intercept.

Since terms of a l inear equations cannotcontain products of distinct or equalvariables, nor any power (other than 1) orother function of a variable, equationsinvolving terms such as xy, x2, y1/3, andsin(x) are nonlinear.


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Solution of Linear Equation:Solution of Linear Equation:--

In mathematics, a system of linear equations(or linear system) is a collection of linearequations involving the same set of

variables. For example,3x + 2y - z = 1

2x - 2y + 4z = -2

-x + y - z = 0

is a system of three equations in the three

variables x, y, z. A solution to a linearsystem is an assignment of numbers to thevariables such that all the equations aresimultaneously satisfied. A solution to thesystem above is given by

x = 1

y = -2


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z = -2

since it makes all three equations valid.

Methods for Finding out TheMethods for Finding out TheSolution of Pair of LinearSolution of Pair of LinearEquations:Equations:

There are mainly two methods for solving outthe solutions of pair of l inear equations,namely:

1) Cramers Rule2) Matrix Method

[1] Cramers Rule:[1] Cramers Rule:

In linear algebra, Cramer's rule is a theorem,which gives an expression for the solution ofa system of linear equations with as many

equations as unknowns, valid in those caseswhere there is a unique solution. The


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solution is expressed in terms of thedeterminants of the (square) coefficientmatrix and of matrices obtained from it by

replacing one column by the vector of righthand sides of the equations. It is named afterGabriel Cramer (17041752), who publishedthe rule in his 1750 Introduction l'analysedes lignes courbes algbriques, althoughColin Maclaurin also published the method inhis 1748 Treatise of Geometry (and probably

knew of the method as early as 1729).The General Case: -

Consider a system of linear equationsrepresented in matrix multiplication form asfollows:

where the square matrix A is invertible andthe vector is the column vector ofthe variables.

Then the theorem states that:

where Ai is the matrix formed by replacingthe ith column of A by the column vector b.


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The rule holds for systems of equations withcoefficients and unknowns in any field , not

just in the real numbers . This formula is,

however, of limited practical value for largermatrices, as there are other more efficientways of solving systems of linear equations,such as by Gauss elimination or, even better,LU decomposition .

Cramers Rule Proof: -Cramers Rule Proof: -

Cramers rule can be proven using two properties of determinants only. The first property being that adding one column toanother doesnt change the value of thedeterminant, and the second property being

that multiplying every element of onecolumn by a factor will increase the value ofthe determinant by the same factor.

Given n linear equations with n variables.

Cramer's rule gives, for the value of x1 theexpression:


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which, using the aforementioned propertiesof determinants can be checked to be true.

In fact, from the equations of the system thisquotient is equal to

By subtracting from the first column thesecond multiplied by x2, the third columnmultiplied by x3, and so on until the lastcolumn multiplied by xn, it is found to beequal to


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,

and according to the remaining property ofdeterminants this is equal to

.

In the same way, if the columns of b's isreplacing the k-th column of the matrix ofthe system of equations the result will beequal to xk. As a result we get that


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Example of Cramers Rule: -Example of Cramers Rule: -

Cramer's rule is an explicit formula for thesolution of a system of linear equations, with

each variable given by a quotient of twodeterminants . For example, the solution tothe system

x + 3y - 2z = 5

3x + 5y + 6z = 7

2x + 4y + 3z = 8


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is given by

For each variable, the denominator is thedeterminant of the matrix of coefficients,while the numerator is the determinant of amatrix in which one column has beenreplaced by the vector of constant terms.

Though Cramer's rule is important theoretically, it has little practical value forlarge matrices, since the computation oflarge determinants is somewhat

cumbersome. (Indeed, large determinantsare most easily computed using rowreduction.) Further, Cramer's rule has very

poor numerical properties, making it unsuitable for solving even small systemsreliably, unless the operations are performedin rational arithmetic with unbounded

precision.


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[2] Matrix Method:[2] Matrix Method:

Let us express the system of linear equationsas matrix equations and solve them usinginverse of the coefficient matrix.

Consider the system of equations

Let

Then, the system of equations can be writtenas, AX = B, i.e.,


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Case I

If A is a non-singular matrix, and then itsinverse exists. Now

AX = B

Or A1 (AX) = A1 B (pre multiplyingby A1)

Or (A1A) X = A1 B (by associativeproperty)

Or I X = A1 B

Or X = A1 B

This matrix equation provides uniquesolution for the given system of equations asinverse of a matrix is unique. This method ofsolving system of equations is known asMatrix Method.

Case II


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If A is a singular matrix, then |A| = 0.

In this case, we calculate (adj A) B.

If (adj A) B O, (O being zero matrix), thensolution does not exist and the system ofequations is called inconsistent.

If (adj A) B = O, then system may be eitherconsistent or inconsistent according as thesystem have either infinitely many solutions

or no solution.

Example of Matrix Method:Example of Matrix Method:

Problem: - Solve the followingsystem of equations by matrixmethod.

Solution: - The system of equations canbe written in the form AX = B, where


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We see that

Hence, A is non-singular and so its inverseexists. Now


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BibliographyBibliographyReferences:

1. http://en.wikipedia.org/wiki/Cramers_rule
http://en.wikipedia.org/wiki/Cramers_rulehttp://en.wikipedia.org/wiki/Cramers_rule


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2. http://en.wikipedia.org/wiki/System_of_linear_equati

ons#Cramer.27s_rule

3. Business Mathematics

4. NCERT Mathematics I, Class 12th
http://en.wikipedia.org/wiki/System_of_linear_equations#Cramer.27s_rulehttp://en.wikipedia.org/wiki/System_of_linear_equations#Cramer.27s_rulehttp://en.wikipedia.org/wiki/System_of_linear_equations#Cramer.27s_rulehttp://en.wikipedia.org/wiki/System_of_linear_equations#Cramer.27s_rulehttp://en.wikipedia.org/wiki/System_of_linear_equations#Cramer.27s_rule

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