Gaussian Elimination Method

What is it?

In Mathematics, the Gaussian elimination method is used in Linear Algebra as a row reduction algorithm for solving Linear Systems. All operations are made on the matrix of coefficients.

The objective is to reduce the matrix to a upper triangular matrix, with the first element being equal to $1$. That way, each equation can be rewritten in a way that is possible to solve for each variable. This is done using the Elementary Operations of Matrix.

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How does it work?

Given a linear system, let’s apply the Gaussian elimination method, in a matrix-based way:

$$\{\begin{array}{l}2x+2y=20\\ 3x+y=22\end{array}\text{rewritten as }A=\left(\begin{array}{ccc}2& 2& |20\\ 3& y& |22\end{array}\right)$$

We can apply two Elementary Operations of Matrix, one on each row. The first will transform the first element to be equal to $1$. The second will transform the the matrix to a upper triangular matrix.

$$\begin{array}{rl}& A_{R_1\to R_1/2}=\left(\begin{array}{ccc}1& 1& |10\\ 3& 1& |22\end{array}\right)\\ & \\ & A_{R_2\to R2-3R_1}=\left(\begin{array}{ccc}1& 1& |10\\ 0& -2& |-8\end{array}\right)\end{array}$$

Now, we can write to a linear system of equations again:

$$\{\begin{array}{ll}& x+y=10\\ & y=4\end{array}$$

A simple replacement of $y=4$ on the first equation would reveal that $x=6$.

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Other methods

One could also consider the Gauss-Jordan Elimination Method, which is an improvement over the Gaussian method.