Solved Problems: Gauss-Jordan Elimination for 3x3 Linear Systems

📚 What is Gauss-Jordan Elimination?

Gauss-Jordan elimination is a powerful algorithm in linear algebra used to solve systems of linear equations. It is an extension of Gaussian elimination, with the added step of reducing the matrix to its reduced row echelon form. This method is particularly useful for solving systems with multiple variables and equations.

• 🔍 Definition: Gauss-Jordan elimination is a method to transform a system of linear equations into an equivalent system that is easier to solve by reducing the augmented matrix to reduced row echelon form.
• 💡 Purpose: To find the unique solution (if it exists) to a system of linear equations by systematically eliminating variables.
• 📝 Advantage: The reduced row echelon form directly gives the solution to the system, making it easy to read off the values of the variables.

📜 A Brief History

The method is named after Carl Friedrich Gauss and Wilhelm Jordan, although variations of the technique were known long before their time. Gauss contributed significantly to the development of the algorithm, while Jordan adapted it for use in geodesy. The method became a cornerstone of linear algebra and is still widely used today.

• 🕰️ Origins: Similar techniques date back to ancient China, but Gauss formalized the process for solving astronomical calculations.
• 🌍 Application: Jordan adapted the method to improve surveying and mapping techniques.
• 🎓 Legacy: Gauss-Jordan elimination is a fundamental tool taught in linear algebra courses worldwide.

🔑 Key Principles

The main idea behind Gauss-Jordan elimination is to perform elementary row operations on the augmented matrix of the system until it is in reduced row echelon form. These operations include:

• ➕ Row Swapping: Interchanging two rows of the matrix.
• ➗ Row Scaling: Multiplying a row by a non-zero scalar.
• ➖ Row Addition: Adding a multiple of one row to another row.

The goal is to transform the matrix so that it has leading 1s (pivots) in each row, with zeros above and below each pivot.

Let's consider a general 3x3 system:

$a_{11}x + a_{12}y + a_{13}z = b_1$

$a_{21}x + a_{22}y + a_{23}z = b_2$

$a_{31}x + a_{32}y + a_{33}z = b_3$

The augmented matrix is:

$\begin{bmatrix} a_{11} & a_{12} & a_{13} & | & b_1 \\ a_{21} & a_{22} & a_{23} & | & b_2 \\ a_{31} & a_{32} & a_{33} & | & b_3 \end{bmatrix}$

🧮 Example: Solving a 3x3 System

Solve the following system using Gauss-Jordan elimination:

$2x + y + z = 2$

$x + 3y + z = 5$

$x + y + 5z = -8$

The augmented matrix is:

$\begin{bmatrix} 2 & 1 & 1 & | & 2 \\ 1 & 3 & 1 & | & 5 \\ 1 & 1 & 5 & | & -8 \end{bmatrix}$

1. Swap Row 1 and Row 2:
   $\begin{bmatrix} 1 & 3 & 1 & | & 5 \\ 2 & 1 & 1 & | & 2 \\ 1 & 1 & 5 & | & -8 \end{bmatrix}$
2. Replace Row 2 with Row 2 - 2 * Row 1, and Row 3 with Row 3 - Row 1:
   $\begin{bmatrix} 1 & 3 & 1 & | & 5 \\ 0 & -5 & -1 & | & -8 \\ 0 & -2 & 4 & | & -13 \end{bmatrix}$
3. Scale Row 2 by -1/5:
   $\begin{bmatrix} 1 & 3 & 1 & | & 5 \\ 0 & 1 & 1/5 & | & 8/5 \\ 0 & -2 & 4 & | & -13 \end{bmatrix}$
4. Replace Row 1 with Row 1 - 3 * Row 2, and Row 3 with Row 3 + 2 * Row 2:
   $\begin{bmatrix} 1 & 0 & 2/5 & | & 1/5 \\ 0 & 1 & 1/5 & | & 8/5 \\ 0 & 0 & 22/5 & | & -49/5 \end{bmatrix}$
5. Scale Row 3 by 5/22:
   $\begin{bmatrix} 1 & 0 & 2/5 & | & 1/5 \\ 0 & 1 & 1/5 & | & 8/5 \\ 0 & 0 & 1 & | & -49/22 \end{bmatrix}$
6. Replace Row 1 with Row 1 - (2/5) * Row 3, and Row 2 with Row 2 - (1/5) * Row 3:
   $\begin{bmatrix} 1 & 0 & 0 & | & 11/22 \\ 0 & 1 & 0 & | & 41/22 \\ 0 & 0 & 1 & | & -49/22 \end{bmatrix}$

Thus, $x = \frac{11}{22} = \frac{1}{2}$, $y = \frac{41}{22}$, and $z = -\frac{49}{22}$.

💡 Real-World Examples

Gauss-Jordan elimination is used extensively in various fields:

• 🧪 Science: Solving complex chemical equations and balancing reactions.
• 📈 Economics: Analyzing supply and demand models.
• 💻 Computer Graphics: Transformations and projections in 3D graphics.
• ⚙️ Engineering: Circuit analysis and structural analysis.

✅ Conclusion

Gauss-Jordan elimination provides a systematic way to solve systems of linear equations. While it can be computationally intensive for large systems, it is a fundamental tool in linear algebra with applications in many different fields. Understanding and mastering this technique is essential for anyone working with linear systems.