Gaussian Elimination

I've recently been taking an interest in linear algebra, the following post is me working through the process and attempting to understand the process of Gaussian elimination using the example provided on the wiki page. It also represents a handy way of testing a new blog feature to allow mathematical notation.

So, to start with we have a set of linear equations, that we can represent as a matrix.

We can add any or subtract any one row of the matrix from another row in addition we can multiply the contents of a row before we do the addition or subtraction. So let's take the first row R1 and mutiply it by 1.5 to give:

Then we can then add it from the the row R2 so that the equation and matrix now look like this:

We also add R1 to R3 which we can do without any mutiplication to give us:

Next we take our newly created R2 row and multiply it by 4 to give:

We can then subtract this from R3 to give:

So now we have a definite value for z of -1 so we can now move back up the matrix. We can multiply R3 by 0.5 and add it to R2 and we can subtract R3 from R1:

Finally we can multiply R2 by 2 and then subtract it from R1 :

Then as a final step we multiply the matrix rows so that each element across the diagonal is now 1 to give us the values for x,y and z.

And that is that...