Solving Homogeneous System

A system of linear equations is homogeneous if all of the constant terms are zero. From Solving System of Linear Equations, the general form of homogeneous system of linear equations can be written as:

a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + … + a_1nx_n = 0

a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + … + a_2nx_n = 0

… … … …

a_m1x₁ + a_m2x₂ + a_m3x₃ + … + a_mnx_n = 0

which can be written in matrix form

AX = 0

Every homogeneous system of linear equations has at least one solution, which is X = 0. It is called zero solution or trivial solution.

• If det(A) is non-zero, then it is the unique solution.
• if det(A) is zero, then there is an infinite number of solutions.

An important technique for solving a homogeneous system of linear equations AX = 0 is to form the augmented matrix M = [ A | 0 ] and reduce M to reduced row echelon form.

Solving with MathPACK Solver

To reduce the augmented matrix M to reduced row echelon form, you can use the function rref() in MathPACK Solver.

Function Signature

U = rref(A)

Example

Solve the homogeneous linear system of equations.

x₁ + x₂ – 2x₃ = 0

3x₁ + 2x₂ + 4x₃ = 0

4x₁ + 3x₂ + 2x₃ = 0

From the given system of linear equations, we can write it in the matrix form

AX = 0

Where:

A = [1, 1, -2; 3, 2, 4; 4, 3, 2];

X = [x₁;x₂;x₃];

So, we can create matrix A and construct augmented matrix M by:


»A = [1, 1, -2; 3, 2, 4; 4, 3, 2]; 

»M = zeros(3, 4); 

»M(1:3,1:3) = A;


Reduce M to reduced row echelon form using rref bundled function


»rref(M) 

ans = 

    1 0 8 0  

    0 1 -10 0 

    0 0 0 0 

Rewrite system in the equations form.

x₁ + 0 + 8x₃ = 0

0 + x₂ – 10x₃ = 0

Thus:

x₁ = – 8x₃

x₂ = 10x₃

Let s=x₃, then

x₁ = – 8s

x₂ = 10s

x₃ = s

The solution vector X will be:

X = [-8; 10; 1]s