Developer website: Cohomology.com
The Linear Algebrator is a Mac OS X application for teaching or learning basic linear algebra, taking you step-by-step through both the Gauss-Jordan elimination algorithm for row reducing matrices as well as the Gram-Schmidt orthogonalization algorithm. You are free to perform row or column operations as you like, or just ask "What do I do next?". An extra drawer displays LU or QR factorizations as you go. All computations are done using fractions, and you can copy or paste the matrix in TeX.
Mac OS X