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.

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.

With that many unknowns I am astonished that this completes in reasonable 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

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.