Systems of Linear Equations - Examples & Applications

This section provides practical examples for solving simultaneous linear algebraic equations using both direct methods (Cramer's Rule, Gauss Elimination, Gauss-Jordan, LU Decomposition) and iterative methods (Gauss-Seidel).

Matrix Fundamentals and Cramer's Rule

Basic: Matrix Multiplication

Given the matrices AA and BB, find the product C=ABC = AB.
A=[21−13],B=[4−205]A = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -2 \\ 0 & 5 \end{bmatrix}

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Basic: Solving a 2x2 System with Cramer's Rule

Solve the following system of equations using Cramer's rule:
3x1+2x2=183x_1 + 2x_2 = 18−x1+2x2=2-x_1 + 2x_2 = 2

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Gauss Elimination

Gauss elimination uses forward elimination to reduce the coefficient matrix to an upper triangular form, followed by back substitution to solve for the unknowns.

Intermediate: Naive Gauss Elimination

Solve the following 3×33 \times 3 system using naive Gauss elimination (without pivoting):
3x1−0.1x2−0.2x3=7.853x_1 - 0.1x_2 - 0.2x_3 = 7.850.1x1+7x2−0.3x3=−19.30.1x_1 + 7x_2 - 0.3x_3 = -19.30.3x1−0.2x2+10x3=71.40.3x_1 - 0.2x_2 + 10x_3 = 71.4

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Gauss-Jordan Elimination and Matrix Inversion

Gauss-Jordan elimination normalizes the pivot rows and eliminates unknowns both above and below the pivot, resulting in an identity matrix on the left side. It is also the standard method to find the inverse of a matrix.

Advanced: Matrix Inversion using Gauss-Jordan

Find the inverse of matrix AA:
A=[2145]A = \begin{bmatrix} 2 & 1 \\ 4 & 5 \end{bmatrix}

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Ill-Conditioning

An ill-conditioned system is one where small changes in the coefficients result in large changes in the solution. This is characterized by a determinant close to zero.

Case Study: Ill-Conditioned System

Consider the following system:
x1+2x2=10x_1 + 2x_2 = 10 1.1x1+2x2=10.41.1x_1 + 2x_2 = 10.4
The exact solution is x1=4,x2=3x_1 = 4, x_2 = 3. Analyze what happens to the solution if the right-hand side of the second equation is changed slightly from 10.410.4 to 10.510.5.

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LU Decomposition

LU decomposition factorizes a matrix AA into a lower triangular matrix LL and an upper triangular matrix UU, such that A=LUA = LU. This is particularly efficient when solving systems with the same coefficient matrix but different right-hand side vectors.

Advanced: LU Decomposition (Doolittle's Method)

Find the LU decomposition of the matrix AA:
A=[3−0.1−0.20.17−0.30.3−0.210]A = \begin{bmatrix} 3 & -0.1 & -0.2 \\ 0.1 & 7 & -0.3 \\ 0.3 & -0.2 & 10 \end{bmatrix}

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Iterative Methods: Gauss-Seidel

The Gauss-Seidel method is an iterative technique that solves for each variable sequentially, immediately using the most recently updated values for subsequent calculations.

Intermediate: Gauss-Seidel Iteration

Use the Gauss-Seidel method to solve the system:
3x1−0.1x2−0.2x3=7.853x_1 - 0.1x_2 - 0.2x_3 = 7.850.1x1+7x2−0.3x3=−19.30.1x_1 + 7x_2 - 0.3x_3 = -19.30.3x1−0.2x2+10x3=71.40.3x_1 - 0.2x_2 + 10x_3 = 71.4
Perform two iterations starting with an initial guess of x1=0,x2=0,x3=0x_1 = 0, x_2 = 0, x_3 = 0. Note: The system is already strictly diagonally dominant.

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