How to find multiple groups of solutions that meet the requirements

Question

I want to find multiple nonzero real matrices A, B that satisfy $A.B=\mathbf{0}$ condition.

A = Array[a, {3, 3}];
B = Array[b, {3, 3}];
FindInstance[
 A.B == ConstantArray[0, {3, 3}] && A != ConstantArray[0, {3, 3}] && 
  B != ConstantArray[0, {3, 3}], Flatten[{A, B}], Reals,5]
{{A, B}} /. First[%]

However, the above code can not return the desired results for a long time. What can I do to quickly get multiple sets of matrices that meet the requirements (their elements are preferably rational numbers)?

Answer 1

With that many unknowns I am astonished that this completes in less than HUGE amounts of time.

{{{a,b,c},{d,e,f},{g,h,i}},{{j,k,l},{m,n,o},{p,q,r}}}/.
FindInstance[{{a,b,c},{d,e,f},{g,h,i}}.{{j,k,l},{m,n,o},{p,q,r}}==
{{0,0,0},{0,0,0},{0,0,0}}&&Element[a|b|c|d|e|f|g|h|i|j|k|l|m|n|o|p|q|r,Reals],
{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r},3]

All the results I've seen have rational or integer entries. It does seem to tend towards solutions with one or more zero rows. If a zero row is acceptable then replacing a,b,c with 0,0,0 seems to speed it up significantly. Eliminating the Element[,Reals] speeds it up even more, but that results in a lot of Complex values.