What are the steps to using the Gaussian method? also known as gaussian elimination.

I wish you were more specific on what part you need help with.Wikipedia has an explanation that might help.

The goal in Gaussian elimination is to make a lot of entries of the matrix zero. That way we can solve a system of linear equations, find the inverse or rank of a matrix much easier.

The best way to understand it is to do a few problems. Wiki already has an example. Look at that and see if it helps. Wolfram also has an entry for that with an example.

In summary you start with looking at the first column of the matrix and try to make all entries zero, except for the first entry. To do that you just add multiples of the first row to the other rows and make them all zero. Then you move to the next column and make the third, fourth, etc. entries of the second column zero. Then you go to the third column and make the fourth, fifth, etc. entries of this column zero. And so on.

Let me know which part you don’t understand and I will try to help.

Gaussian elimination is fun. It’s kinda like a puzzle. You want to get your matrix to look like,

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

with any number on the ends of those rows, if I remember correctly. You have the options of doing a few different types of transformations on the numbers in order to systematically achieve that identity matrix.

Then you can solve the problem as a system of linear equations.

The idea behind Gaussian elimination is fairly straightforward. I am going to explain just using words and no diagrams, but I still think it should be fairly clear. If you need further help, I can come up with a specific example and go through it step by step.

Suppose you have a 3 by 3 matrix in variables x, y and z, where x is the first column, y the second and z the third.

Choose a variable, say x, and choose one of the rows where the x coefficient is non-zero. Now subtract that row from each of the other rows having non-zero x coefficient, in such a way that you will end up with zero coefficient for x in the other rows.

Now choose a row with non-zero coefficient for y, but you want it to make sure it is different from the row you chose previously. This means that the row will have a zero x coefficient from the previous step. And similar to what you did before, subtract the row from each other row with non-zero y coefficient, in such a way that you end up with a zero y coefficient in the other rows.

If you think about it for a moment you will see that there will be at most one row with non-zero x coefficient and at most one with non-zero y coefficient.

Suppose that at this point that the third row (the one not chosen in the previous steps) has all zeroes in it, so that you can’t continue, as previously, to choose a row for z different from the two already chosen. This means that you are done. z is the only independent variable with both x and y expressible in terms of z.

I hope this makes sense and that you see how to generalize the method. If there are no degenerate rows that go to all zeroes then you end up with something like what @girlofscience showed. In this case the only solution is having all variables equal to 0.