As I do not own an Nspire, I don't know how. However, I was able to make a TI-Basic program on the 83+/84+ calculators. It uses the matrix [B] as the matrix with your original values. Note, this is not the most optimized, but it should work for any size matrix.
For your starting matrix, just create a matrix in [B] that is however wide by however tall, then enter your values, and then run the program, and the program will show you the common factor (or whatever) multiplied by the new, simplified matrix, like you showed above.
Like I said, I don't know how you would do this on Nspire, but this works for 83+/84+ for sure.