I'm obviously not going to ask for manual G-J elimination of
a big matrix on a test, but I might ask for you to tell me the
steps for something small, like a 2x3 augmented matrix.

In order to practice manual Gauss-Jordan elimination,
I recommend 3x4 or smaller augmented matrices. You
can also practice "semi-automatic" Gauss-Jordan elimination
on slightly bigger matrices without too much trouble.
By this I mean use your calculator's row operation commands
to perform row operations one by one.

http://www2.ohlone.edu/people2/joconnell/ti/matrixops89.pdf

You can also perform row operations one by one at

http://live.sympy.org

Just first copy over my row_op function definition from

http://nbviewer.jupyter.org/urls/dl.dropbox.com/s/j0pmptjt3jkelzc/02-rref.ipynb

and evaluate it. Then you can use row_op like I do in the above link.