The elementary operations are:

- Interchange two of the equations in the system.
- Multiply one of the equations through by a number other than zero.
- Add a multiple of one equation to another.

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.