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Elementary Operations on a System of Linear Equations

There are three basic kinds of operations that can be used to solve any system of linear equations. These operations are simple transformations of the original system of equations that preserve the information content about the solution.

The elementary operations are:

Clearly, these elementary transformations preserve the information content of the original system of equations for the simple reason of being reversible. In other words, after applying a sequence of these operations to the original system we'll end up with another system that may look very different from the initial one, but by applying the reversed sequence in reverse we can go back to the begining. Notice that we undo the first kind of elementary transformation (switching two rows) by doing it again. That is, the reverse of switching is itself!

Let us call k the number that multiplies the equations in the definition of the elementary operations of the second and third kind above. We can now see that the operations that undo them are also of the second and third kind but with -k instead of k.

OK. But what's so special about these operations?
Well, they are enough to solve ALL possible linear systems of equations. We'll see that by applying sequences of these transformations we can always find the solution (if there is a solution) to the system or find out that there is no solution. A systematic procedure to accomplish this is Gaussian Elimination.