Steps for Gauss-Jordan Elimination to Solve Linear Systems

📚 Understanding Gauss-Jordan Elimination

Gauss-Jordan elimination is a powerful algorithm in linear algebra used to solve systems of linear equations, find the inverse of a matrix, and compute determinants. It's an extension of Gaussian elimination, refining the process further to achieve a reduced row echelon form.

📜 History and Background

The method is named after Carl Friedrich Gauss and Wilhelm Jordan, though variations of the technique were known much earlier. Gauss made significant contributions to the method around 1810. Jordan adapted the method for geodetic surveys around 1888.

🔑 Key Principles

The core idea is to transform a given matrix into its reduced row echelon form through a series of elementary row operations. This makes solving systems of equations straightforward.

🪜 Steps for Gauss-Jordan Elimination

• 🔢 Represent the system as an augmented matrix: Write the coefficients of the variables and the constants as an augmented matrix.
• 🎯 Forward Elimination (Gaussian Elimination):
  • 🥇 Find the pivot: Select the first non-zero entry in the first column (the pivot).
  • ➗ Normalize the pivot: Divide the pivot row by the pivot to make the pivot entry equal to 1.
  • ➖ Eliminate entries below the pivot: Subtract multiples of the pivot row from the rows below it, to make the entries in the column below the pivot equal to 0.
  • ♻️ Repeat: Repeat these steps for the next column, ignoring the rows above the current pivot.
• ⏪ Backward Elimination (Jordan's Contribution):
  • ⬆️ Eliminate entries above the pivots: Starting from the last pivot, subtract multiples of the pivot row from the rows above it to make the entries in the column above the pivot equal to 0.
• ✅ Check for Reduced Row Echelon Form: Ensure the matrix is in reduced row echelon form: all pivots are 1, entries above and below pivots are 0, rows of all zeros are at the bottom.
• 🎉 Read the Solution: The rightmost column of the reduced row echelon form represents the solution to the system of equations.

🧪 Example: Solving a System of Equations

Let's solve the following system using Gauss-Jordan elimination:

$\begin{cases} x + y + 2z = 9 \\ 2x + 4y - 3z = 1 \\ 3x + 6y - 5z = 0 \end{cases}$

1. Augmented Matrix: $\begin{bmatrix} 1 & 1 & 2 & 9 \\ 2 & 4 & -3 & 1 \\ 3 & 6 & -5 & 0 \end{bmatrix}$
2. Forward Elimination:
   • Row 2 = Row 2 - 2 * Row 1: $\begin{bmatrix} 1 & 1 & 2 & 9 \\ 0 & 2 & -7 & -17 \\ 3 & 6 & -5 & 0 \end{bmatrix}$
   • Row 3 = Row 3 - 3 * Row 1: $\begin{bmatrix} 1 & 1 & 2 & 9 \\ 0 & 2 & -7 & -17 \\ 0 & 3 & -11 & -27 \end{bmatrix}$
   • Row 2 = Row 2 / 2: $\begin{bmatrix} 1 & 1 & 2 & 9 \\ 0 & 1 & -\frac{7}{2} & -\frac{17}{2} \\ 0 & 3 & -11 & -27 \end{bmatrix}$
   • Row 3 = Row 3 - 3 * Row 2: $\begin{bmatrix} 1 & 1 & 2 & 9 \\ 0 & 1 & -\frac{7}{2} & -\frac{17}{2} \\ 0 & 0 & -\frac{1}{2} & -\frac{3}{2} \end{bmatrix}$
   • Row 3 = Row 3 * -2: $\begin{bmatrix} 1 & 1 & 2 & 9 \\ 0 & 1 & -\frac{7}{2} & -\frac{17}{2} \\ 0 & 0 & 1 & 3 \end{bmatrix}$
3. Backward Elimination:
   • Row 1 = Row 1 - 2 * Row 3: $\begin{bmatrix} 1 & 1 & 0 & 3 \\ 0 & 1 & -\frac{7}{2} & -\frac{17}{2} \\ 0 & 0 & 1 & 3 \end{bmatrix}$
   • Row 2 = Row 2 + \frac{7}{2} * Row 3: $\begin{bmatrix} 1 & 1 & 0 & 3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{bmatrix}$
   • Row 1 = Row 1 - Row 2: $\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{bmatrix}$

Therefore, $x = 1$, $y = 2$, and $z = 3$.

📊 Real-World Applications

• 🌍 Economics: Solving systems of equations in economic models.
• ⚙️ Engineering: Circuit analysis and structural analysis.
• 📈 Computer Graphics: Transformations and projections in 3D graphics.

💡 Tips for Success

• 📝 Organization: Keep your matrix organized to avoid errors.
• ➕ Careful Arithmetic: Double-check your calculations at each step.
• practice Practice: The more you practice, the better you'll become at recognizing patterns and avoiding mistakes.

🎓 Conclusion

Gauss-Jordan elimination is a fundamental tool in linear algebra. By mastering the steps, you can confidently solve systems of linear equations and apply this technique to various real-world problems.