Pairs of nonsingular matrices

Let Plt llln denote the se t of m by n matrices of rank In, with e ntries from a field of characteri sti c O. Given REPltmn let .IV (R) denote the set of 5EPltnm , n such that R5T = O. Note thal 5E .IV(R) if and only if RE.!V (5) , We assume as known the following results for REfYtmn and 5E.IV (R): LEMMA 1. XR = 0 if and only if X = O. LEMMA 2. RYT = 0 if and only if Y = ZS , for some matrix Z. The obvious stipulation s are that X, Y, and Z have m,n and n-In column s respec tively, and Y and Z have the same number of rows . Using these lemmas we prove: THEOREM. If R" R2EPltmn ' StEff (R,) , S2Eff (R2) then R,RJ is nonsingular if and only if SISi' LS nonsingular, in which case