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Index
See also: Gauss-Jordan 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.

Definition: Row Equivalence Operations
 
Let A be an arbitrary matrix. Then the following operations are called row equivalence operations:
 
  • Swap two rows of A
  • Multiply any row of A by any number not equal to 0
  • Replace a row of A by the sum of this row, and another row, multiplied by any number (including 0)


All of these 3 operations keep all essential properties of a matrix. Foremost, they don't change the solution of the system of linear equations depicted by A.

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.
 
 

Example: Calculating the inverse of a matrix manually

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
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
1
0		 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 sub-matrix.

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: 2004-Jul-03