Diagonal matrices have relatively few distinct minors. Applying `rcop' one or more times to a diagonal matrix generates equivalent matrices that, in most cases, will have more distinct minors. (These operations do not, of course, change the greatest common divisor of the kxk minors for any k.)
The game is to try to find a matrix equivalent to a given diagonal matrix for which the kxk minors are all different. For example, starting with a diagonal 3x4 matrix in which the diagonal entries are all nonzero, try to perform rcops in order to find equivalent matrices that have more and more distinct kxk minors for each k. You should, in fact, be able to find such matrices in which no two kxk minors are the same, that is, there are 12 different entries, 18 different 2x2 minors, and 4 different 3x3 minors.
Note that even when there are lots of different kxk minors it is easy to see what their greatest common divisor is and therefore to know the product of the first k diagonal entries of the equivalent strongly diagonal matrix. From this, the strongly diagonal matrix itself is easy to deduce.
Return to Chapter 5.