  We introduce the Bhattacharya-Mesner rank of third order hypermatrices as a
relaxation to the tensor rank and devise from it some bounds for the tensor
rank. We use the Bhattacharya-Mesner rank to extend to third order
hypermatrices the connection relating the rank to a notion of linear
dependence. We also derive explicit necessary and sufficient conditions for the
existence of third order hypermatrix inverse pair. Finally we use inverse pair
to extend to third order hypermatrices the formulation and proof of the matrix
rank-nullity theorem.
