2. The adjoint of a matrix, A, can be defined as the matrix such that $(Au)\cdot v= u \cdot(adj A v)$.
If A has rank n- 1, then it had "nullity" n- (n-1)= 1. That is, the set of all vectors u, such that Au= 0, is a one dimensional subspace. In particular, for any such u, $(Au)\cdot v= 0= u\cdot(adj A v)$.