Elementary operations

What can we do to this matrix in order to solve the equations? We can interchange any two equations at a time. We will call the resulting operation if it exchanges the -th and the -th equation. In Maple, this can be accomplished by the swaprow command.

We can also multiply the -th equation by a number and add it to the -th equation. We shall call this operation . In Maple, this can be accomplished by the command addrow.

We can also multiply the -th equation by a nonzero number k. We will accomplish this by a command we made up, called multrow . We may denote it by .

For example, we illustrate a sequence of these operations which will be useful for solving the equations. We explain the steps later.

Operation .

> M1:=swaprow(M,1,2);

Operation .

> M2:=addrow(M1,1,3,-2);

Operation .

> M3:=addrow(M2,2,3,-3);

Operation .

> M4:=multrow(M3,3,-1);

>

These last equations are in a particularly nice form, if we think of them from bottom to top. The third, one solves for , then the second can be solved for , since it only involves and the first one can then be solved for . We could also do more operations which reflect the same work, but in a compact form!

> M5:=addrow(M4,3,2,-2);

> M6:=addrow(M5,3,1,-3);

> M7:=addrow(M6,2,1,1);

These are clearly solved, giving , and .

Our aim is to understand and generalize this process. There are two simple maple commands which can readily give the stages M4 and M7 immediately. We wish to master them!

> gausselim(M),gaussjord(M);

>