The Linear Algebrator
 
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.
Download Linear Algebrator 1.0 (August 4, 2006)
The Linear Algebrator is shareware. This download is fully functional, and you are welcome to try it before purchasing. However, if you use it for more than a week, use it in an in-class demonstration, or it helps you with your homework or exams, you must purchase a US$10 registration via the “Purchase Registration...” menu item.
If you have any questions, comments, or suggestions, please e-mail Greg Landweber.
Since a picture is worth a thousand words, here is a screenshot. I have just asked “What do I do next?”.
I have just orthogonalized the above matrix. Note the use of fractions and the QR factorization drawer.