This program starts with a random matrix, and lets you apply some extremely elementary row operations, and also the transpose operation. You can do a lot with this small selection of operations: you should be able to make it upper triangular without using the transpose, and diagonal with the help of the transpose.
If you ask for a new matrix, it uses random entries between -2 and 2. This could be made more interactive if anybody is interested, or indeed if there is anybody out there at all.