Can you describe how Gaussian elimination works?

Gaussian elimination, also known as row reduction, is a method for solving linear systems of equations by transforming a given matrix (augmented matrix) to row echelon form using a series of elementary row operations. Once the matrix is in row echelon form, we can then use back-substitution to find the unique solution (if it exists). There are three possible outcomes when using Gaussian elimination: a unique solution, infinitely many solutions, or no solution (i.e., an inconsistent system).

Here is an overview of the Gaussian elimination algorithm:

1. Starting with the augmented matrix for the linear system of equations.

2. Perform elementary row operations (ERO) to get a matrix in row-echelon form. ERO includes:

(a) Swap two rows.

(b) Multiply a row by a non-zero scalar.

(c) Add or subtract a multiple of a row from another row.

3. Once in row-echelon form, perform back-substitution to solve for the variables.

Let’s work through an example to illustrate Gaussian elimination:

Consider the following system of linear equations:

x - 2y +  z =  1
2x +  y +  z =  0
-3x - 4y + 2z = 6 

First, write down the augmented matrix:

A = |  1 -2  1 |  1 |
    |  2  1  1 |  0 |
    | -3 -4  2 |  6 |

To get the row-echelon form, we first find the pivot in the first row (1) and use it to eliminate the elements below the pivot.

Row2 = Row2 - 2 * Row1
Row3 = Row3 + 3 * Row1

A = |  1 -2  1 |  1 |
    |  0  5 -1 | -2 |
    |  0 -2  5 |  9 |

The pivot in the second row is 5. Use it to eliminate the element below it in row 3.

Row3 = Row3 + 2/5 * Row2

A = |  1 -2   1   |  1 |
    |  0  5  -1   | -2 |
    |  0  0   9/5 |  4 |

Now we have reached the row-echelon form of the matrix. We can use back-substitution to solve for the variables:

9/5 z = 4
z = 20/9

5y - z = -2
y = -2/5 + (20/45)
y = -1/9

x - 2y + z = 1
x = 1 + 2 * 1/9 - 20/9
x = 8/9

So, the solution to the system of equations is ‘x = 8/9‘, ‘y = -1/9‘, and ‘z = 20/9‘.

Gaussian elimination is widely used in mathematical finance, particularly when solving systems of linear equations that arise in portfolio optimization, linear regression, and other optimization problems. The method can be extended to solve larger systems of equations and can be efficiently implemented in computer programs.