Do any such programs exist?
You mean you're looking for a free program that can work with matrices??
Sure, that exists. Here is one: http://maxima.sourceforge.net/
I second DaleSpam, Python and http://numpy.scipy.org/" [Broken] is what you want. (Maxima is good with numerics, but is more known for its symbolics)
However, if you're used to something like matlab, then have a look at http://www.gnu.org/software/octave/doc/interpreter/Introduction.html#Introduction".
Also, most programming languages have (free) libraries for array/matrix operations / linear algebra.
E.g. ATL or uBLAS are the standards for C++. Just google for more.
Separate names with a comma.