Solving Systems of Linear Equations

Topics: System of linear equations, Linear algebra, Elementary algebra Pages: 30 (3080 words) Published: March 11, 2013
Solving systems of linear equations

7.1 Introduction
Let a system of linear equations of the following form:

a11 x1
a21 x1

a12 x2
a22 x2

ai1x1  ai 2 x2

am1 x1  am2 x2


a1n xn
a2 n x n

   ain xn

   amn xn



 bm


be considered, where x1 , x2 , ... , xn are the unknowns, elements aik (i = 1, 2, ..., m; k = 1, 2, ..., n) are the coefficients, bi (i = 1, 2, ..., m) are the free terms of the system. In matrix notation, this system has the form:

Ax  b ,


where A is the matrix of coefficients of the system (the main matrix), A = [aik]mn, b is the column vector of the free terms, bT  [b1 , b2 , ... , bm ] , x is the column vector of the unknowns, xT  [ x1 , x2 , ... , xn ] ; the symbol () T denotes transposition.

It is assumed that aik and bi are known numbers. An ordered set, {x1, x2, ..., xn}, of real numbers satisfying (7.1) is referred to as the solution of the system, and the individual numbers, x1, x2, ..., xn, are roots of the system.

A system of linear equations is:

consistent - if it has at least one solution. At the same time it can be -

determined - if it has exactly one, unique solution,
undetermined - if it has infinitely many solutions;

inconsistent - if it does not have any solution.

The further considerations will be limited to most frequently met in technical problems non-homogeneous systems of equations, i.e., systems that satisfy:




 0.


i 1

Conditions for existing of solution of such systems give the following theorem:

Theorem 7.1 (Kronecker-Capelly)
The system of equations (7.1) is solvable if and only if, in notation (7.2):

rank(A) = rank(B) = r,

where B is the extended matrix created by inserting vector b as the n+1 column of matrix A. At the same time:

there exists exactly one, unique solution if r = n;

there are infinitely many solutions dependently on the number of parameters, e = n - r, if r < n;


the system is inconsistent if rank(A) < rank(B).


Common rank, r, of matrices A and B is referred to as the system order. Calculation of the solution of a non-homogenous system of linear equations involves:

1. Determining the system order, r, and respective non-zero minor, D, of matrix A.

2. Selecting equations, whose coefficients are present in minor D and rejecting other equations. In the selected equations r unknowns at coefficients creating the minor D are left on the left-hand side of the r equations, and the remaining n - r unknowns are considered as parameters and left on the right-hand side of the equations.

3. Solving the obtained system of r equations with r unknowns. The unknowns are linear functions of n - r parameters. The solution is called the general solution.

4. Assuming arbitrary values for the parameters. The solution obtained is called the particular solution.

The above procedure requires skills of solving n-th (or r-th, if r < n) order systems of linear equations. Some methods, both accurate and approximate, are presented in following sections.


7.2 Accurate methods
The term accurate methods (referred to also as direct methods) will be used for methods of solving systems of linear equations, which allow finding accurate solutions in a finite number of elementary arithmetical operations, where this number depends on the algorithms used and the system order.

7.2.1 Cramer formulae
A Cramer system, i.e., system of n equations in n unknowns, is given by (7.2). Provided the matrix, A, of coefficients is non-singular, i.e., det A  0 , there exists an inverse matrix A-1. In order to find the solution of the system, both sides of (7.2) are premultiplied by A-1, what gives:

x  A1b .


The inverse matrix is calculated from the following equation:

A 1 

det A

where AD is the transposed matrix of...
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