The range of the iterated matrix adjoint operator

The following inverse problem is considered: for a given n × n real matrix B, does there exist a real matrix A such that where the classical adjoint operation is intended? The rank of B and the number of applications of the adjoint operator determine the character of this general inverse problem for the iterated adjoint operator. Thus, for given B, the question of interest is whether or not B lies in the range of the iterated matrix adjoint operator. Maple V R5 is used as an aid to obtain results indicated here.