Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more. 
See also: GaussJordan algorithm, linear equations  
Equivalence Operations
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.
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.
An example of how to take advantage by either row or column equivalence operations is the manual calculation of an inverse. Let's calculate the inverse of
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).
Next, we add two times the second row to the first row (rule 2) and
get
After finally having multiplied the second row with 1 (rule 3), we
end up with:
So, we eventually have calculated A^{1}:


Last Update: 20121008