Let F be the set of pairs of matrices (A,B)∈M2(Z)×M2(Z) for which there exists some positive integer k and matrices C1,C2,…,Ck∈{A,B} such that C1C2⋯Ck=O2. For each (A,B)∈F, let k(A,B) denote the minimal positive integer k which satisfies the latter property.[*]Let (A,B)∈F with det(A)=0,det(B)=0 and k(A,B)=p+2 for some p∈N∗. Show that ABpA=O2.
[*]Prove that for any k≥3 there exists a pair (A,B)∈F such that k(A,B)=k.
Bogdan Blaga