On sparse random combinatorial matrices

Let Qn,d denote the random combinatorial matrix whose rows are independent of one another and such that each row is sampled uniformly at from subset vectors in {0,1}n having precisely d entries equal to 1. We present a short proof fact P[det?(Qn,d)=0]=O(n1/2log3/2?nd)=o(1), whenever ?(n1/2log3/2?n)=d?n/2. In particular, our accommodates sparse matrices sense d=o(n) allowed. also consider singularity deterministic integer A randomly perturbed by matrix. we prove P[det?(A+Qn,d)=0]=O(n1/2log3/2?nd), again, ?(n1/2log3/2?n)=d?n/2 has property (1,?d) not an eigenpair A.