Julia, 12

There are not too many valid pairs of them but there are ten distributed over the dimensions for base 10.

Julia, 9

There are not too many valid pairs of them but there are ten distributed over the dimensions for base 10. It also produces only a single 12x12 solution therefore it was agreed to count the method at the level of the second largest solutions it produces, ie 9.

I do a random search for nxn-matrices of the form

fulfilling the criteria. As these are full, I can make block-matrices out of two of them to get a half-filled solution.

I identify a class of full rank matrices for which the criterion A*B = 10A + B can be checked in constant time, nxn-matrices of the form

Then I do a random search for those matrices in this class fulfilling the criteria. As these are full, I can make block-matrices out of two random elements of the class get a half-filled larger solution.

There are not too many of them but more than ten distributed over the dimensions for base 10.

There are not too many valid pairs of them but there are ten distributed over the dimensions for base 10.