Gaussian Elimination

What is Gaussian Elimination?

Gaussian elimination is a method for solving systems of linear equations. It is named after Carl Friedrich Gauss, although it was known to Chinese mathematicians as early as 179 AD. This method uses elementary row operations to modify the coefficient matrix of the system into row echelon form, from which the solution can be obtained by back-substitution.

The Process of Gaussian Elimination

For a system of linear equations \(AX = B\), where A is the coefficient matrix, X is the variable vector, and B is the constant vector, we follow these steps:

1. Write the augmented matrix \([A|B]\).
2. Convert the augmented matrix to row echelon form using elementary row operations:
   • Swap rows (if necessary)
   • Multiply a row by a non-zero number
   • Add a multiple of one row to another
3. Perform back-substitution to solve for the variables.

Example Calculation

Let's solve the following system of equations:

\[ \begin{cases} 2x + y - z = 8 \\ -3x - y + 2z = -11 \\ -2x + y + 2z = -3 \end{cases} \]

1. Augmented matrix: \[ \begin{bmatrix} 2 & 1 & -1 & 8 \\ -3 & -1 & 2 & -11 \\ -2 & 1 & 2 & -3 \end{bmatrix} \]
2. Row echelon form (after elementary row operations): \[ \begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{bmatrix} \]
3. Solution: \[ \begin{cases} x = 2 \\ y = 3 \\ z = -1 \end{cases} \]

Visual Representation

This diagram shows the solution point (2, 3, -1) in the XY plane. The Z-coordinate is represented by the size of the point.