Matrix | Tensor Analysis Answers

Suppose nxn martix A is non-negative (most entries are positive). Suppose M is an nxn projection matrix with rank m>0. How can I show (or derive a minimal condition for) that tr(AM) is strictly positive? Thanks!

Prove that tr(A.A[sup]T[/sup])=tr(A[sup]T[/sup].A) for every matrix A.

Prove that the trace of a matrix A i.e TR(A)=sum of the diagonal elements of A.