## Simple Matrix Calculator

This will take a matrix, of size up to 5x6, to reduced row echelon form by
Gaussian elimination. Each elementary row operation
will be printed. Given a matrix smaller than 5x6, place it
in the upper lefthand corner and leave the extra
rows and columns blank. Some sample values have been included. Press
"Clear" to get rid of them.

A is a 2x2 matrix and B is 2x1 matrix.
This calculator will attempt to find AB and solve
AX=B by calculating A^{-1}B, when possible.
Otherwise it will report whether it is consistent.
It also gives det, rank and eigenvalues.

A few comments about what's going behind the scenes:
The Gaussian elimination uses the same
algorithm that you would use by hand: work left to
right, and do elementary row operations to get each column to
have the right shape.
This puts the matrix in echelon
form. Then go back and eliminate extra entries to get
it into reduced form.

For the second part, the inverse is computed by the
standard formula using determinants
A^{-1} = (1/det(A))adjoint(A)
This works great in our (2x2) case, although it is not feasable
for larger matrices. The consistency test is based on idea that
for AX=B to be solvable, we need B to be in the
column space of A. So it suffices to check that the
dimension of Col(A) = Col(A|B). That is that the
ranks of A and (A|B) are the same. The ranks are
computed using minors.
The eigenvalues are gotten by solving the
characteristic equation
det(tI-A) = t^{2} -trace(A)t+det(A) =0
Since it's quadratic, it's easy to do this explicitly.
This essentially a "demo" which I put together for
teaching purposes. It should be fine for simple examples
but it can occasionally give wrong answers due
to **round off errors**.
It will usually try to warn you when such errors are likely.
There is plenty of serious matrix crunching software
out there. Some of it free (octave or
scilab) and some not
( matlab ).

- Donu Arapura