Pairs of maps preserving singularity on subsets of matrix algebras

Let F be an algebraically closed field and M_n be the n×n matrix algebra over F. A total graph of the full matrix algebra is the graph with M_n as vertices, and two distinct matrices A,B are adjacent if and only if A+B is singular. The characterization of all the automorphisms of the total graph is an open question. Motivated by this problem, we study pairs of maps on a subset of M_n preserving the singularity of matrix pencils A+λB. In particular, we characterize maps T₁,T₂:M_n→M_n satisfying the condition A+λB is singular if and only if T₁(A)+λT₂(B) is singular, for any A,B∈M_n and any non-zero λ∈F. Namely, we prove that in this case T₁=T₂ and they are of the form T₁(A)=T₂(A)=PAQ for all A∈M_n, or of the form T₁(A)=T₂(A)=PA^tQ for all A∈M_n, where P,Q∈M_n are non-singular matrices.