Gauss Method

Description

The Gauss elimination is a method for solving systems of linear equations of $n$ equations and $n$ unknowns:

\begin{matrix} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1 \\ a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2 \\ \vdots & & \vdots & & & & \vdots & = & \vdots \\ a_{n1}x_1 & + & a_{n2}x_2 & + & \cdots & + & a_{nn}x_n & = & b_n \end{matrix}

Represent the system as an augmented matrix
Manipulate the rows to form an upper triangle matrix by:
- Swapping the positions of two rows.
- Multiplying a row by a non-zero scalar.
- Adding to one row a scalar multiple of another.
Starting from the last equation (which is solved), substitute the value on the others to get the value of the other unknowns.

Number of equations

a11 x1

a12 x2

a13 x3

= b1

a21 x1

a22 x2

a23 x3

= b2

a31 x1

a32 x2

a33 x3

= b3