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See also: GaussJordan algorithm, linear equations 
A key to solving linear equations is equivalence operations which change a system of linear equations without changing the solution. For the following section it is important to remember that systems of linear equations can be depicted as matrices.
Definition: Row Equivalence Operations
Let A be an arbitrary matrix. Then the following operations
are called row equivalence operations:

For other purposes, you may replace row operations by column operations
leading to the same result, though they don't keep the solutions of a system
of linear equations.
A :=  0
1 
1
2 
First of all, we extend this matrix with the identity matrix of appropriate
order (red part):
0
1 
1
2 

0
1 
Now we are going to apply equivalence operations to transform the green part of the extended matrix into an identity matrix. This will result in the inverted matrix A^{1} contained in the red submatrix.
We start by swapping both rows (rule 1 of those mentioned above).
1
0 
2
1 
0
1 
1
0 
Next, we add two times the second row to the first row (rule 2) and
get
1
0 
0
1 
2
1 
1
0 
After finally having multiplied the second row with 1 (rule 3), we
end up with:
1
0 
0
1 
2
1 
1
0 
So, we eventually have calculated A^{1}:
A^{1} =  2
1 
1
0 
Last Update: 2004Jul03