# What is Gauss-Jordan elimination?

Oct 30, 2014

Gauss-Jordan elimination is a technique for solving a system of linear equations using matrices and three row operations:

1. Switch rows
2. Multiply a row by a constant
3. Add a multiple of a row to another

Let us solve the following system of linear equations.

$\left\{\begin{matrix}3 x + y = 7 \\ x + 2 y = - 1\end{matrix}\right.$

by turning the system into the following matrix.

$R i g h t a r r o w \left(\begin{matrix}3 \text{ "1" "" "7 \\ 1" "2" } - 1\end{matrix}\right)$

by switching Row 1 and Row 2,

$R i g h t a r r o w \left(\begin{matrix}1 \text{ "2" "-1 \\ 3" "1" "" } 7\end{matrix}\right)$

by multiplying Row 1 by -3 and add it to Row 2,

$R i g h t a r r o w \left(\begin{matrix}1 \text{ "" "2" "-1 \\ 0" "-5" } 10\end{matrix}\right)$

by multiplying Row 2 by $- \frac{1}{5}$,

$R i g h t a r r o w \left(\begin{matrix}1 \text{ "2" "-1 \\ 0" "1" } - 2\end{matrix}\right)$

by multiplying Row 2 by -2 and add it to Row 1,

$R i g h t a r r o w \left(\begin{matrix}1 \text{ "0" "" "3 \\ 0" "1" } - 2\end{matrix}\right)$

by turning back into a system of equations,

$R i g h t a r r o w \left\{\begin{matrix}x = 3 \\ y = - 2\end{matrix}\right.$,

which is the solution of the original system.

I hope that this was helpful.