RREF for 3-Variable Equations with Geometric Ratios

Hello everyone, I'll start off my first post by explaining a few things.

The equations used in this matrix all have their coefficients related by a common ratio, for example:

8x+4y+2z=1 all have a ratio of 2 and so forth.

When I have three of such equations I find that the there is always a definite solution to these equations. With its appearance being that of an identity matrix I was wondering if there were any pattern related to the solutions as I found one for a 2x2 matrix.

Here are my algebraic expressions:

ax+(ar)y+(ar^2)z=ar^3

bx+(bn)y+(bn^2)z=bn^3

jx+(jk)y+(jk^2)z=jk^3

where a,b and j are not equal, and r/n/k are ratios respectively.

I put this into a matrix in its condensed form

Could anybody help me with using the Guassian method of elimination of help me obtain a resulting solution where it will look like a 3x3 identity matrix.

Thanks. (Cool)