Linalg_GaussJordan.html

Solutions of Linear Systems - Gauss Jordan Method

Linear Algebra Project

Objectives:

To investigate solutions of systems of linear equations by

(a) Matrix Method

(b) Step-by-step Gauss-Jordan Elimination

These steps will be carried out using Maple commands.

(d) To discuss the solution set of a system by graphing whenever possible.

Solved Example 1:

Solve: 4x + 3y = 2

7x + 5y = 3

Solution:

(a) Matrix Method

First activate the linalg and plot packages of Maple.

> with(linalg): with(plots):

Warning, new definition for norm

Warning, new definition for trace

Matrix Method: Define the coefficient matrix: A:= matrix(2,2,[4,3,7,5]).

Define the constant matrix (or vector): c:= vector([2, 3]);

> A:= matrix(2,2,[4,3,7,5]);

> c:= vector([2,3]);

Use the command "linsolve" to solve the system.

> linsolve(A,c);

(b) Gaussian Elimination - Step by Step:

First define the augmented matrix.

> Ac:= augment(A,c);

Now we use the three elementary row operations to get an echelon

> A1:= mulrow(Ac, 1, 1/4);

> A2:= addrow(A1,1,2,-7);

> A3:= addrow(A2,2,1,3);

> A4:=mulrow(A3,2,-4);

The solution is x = -1, y = 2.

(c) Gaussian Elimination - in one step.

> Acr:=rref(Ac);

This operation reduces the matrix to r ow r educed e chelon f orm (rref) and basically solves the problem.

rref is also called "reduced normal form."

(d) Graph of the system

We use the command "implicitplot".

> implicitplot({4*x + 3*y = 2,7*x + 5*y = 3}, x =-1.5..-.5, y = 1..3);

Solved Example 2:

Solve the linear system

x + 2y + 3z = 0

x+ 3y + 8z = 2

x + 2y + 2z = 0

by Gauss-Jordan method.

Solution:

First define the augmented matrix then use RREF command.

> A:= matrix(3,3,[1,2,3,1,3,8,1,2,2]);

A := matrix([[1, 2, 3], [1, 3, 8], [1, 2, 2]])

> d:= vector([0, 2, 0]);

> Ad:= augment(A,d);

Ad := matrix([[1, 2, 3, 0], [1, 3, 8, 2], [1, 2, 2, 0]])

> Adr:= rref(Ad);

Solution: x = -4, y = 2, z = 0.

Graph of the system

Geometrically, each linear equation in three variables is a plane. To find the solution of the system is to find the point where the three planes intersect each other.

We use the "implicitplot3d" command to graph an equation in three variables.

> implicitplot3d({x + 2*y + 3*z = 0,x+ 3*y + 8*z = 2,x + 2*y + 2*z = 0},x=-5..-3,y=1..3, z= -1..1, title=`Graph of the System`);

We can get some idea about the location where the three planes meet by rotating the box with y-axis backward and forward, x-axis left and right, z-axis up and down, and noting that the planes intersect on the lines through x = -4, y = 2 and z = 0.

We zoom repeatedly to get the exact solution. We can find the exact location by rotating the axes.

> implicitplot3d({x + 2*y + 3*z = 0,x+ 3*y + 8*z = 2,x + 2*y + 2*z = 0},x=-4.0001..-3.9999 ,y=1.9999..2.0001, z= -0.0001..0.0001, title=`Graph of the System`);