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We introduce the Bhattacharya-Mesner rank of third order hypermatrices as arelaxation to the tensor rank and devise from it some bounds for the tensorrank. We use the Bhattacharya-Mesner rank to extend to third orderhypermatrices the connection relating the rank to a notion of lineardependence. We also derive explicit necessary and sufficient conditions for theexistence of third order hypermatrix inverse pair. Finally we use inverse pairto extend to third order hypermatrices the formulation and proof of the matrixrank-nullity theorem.

On the Bhattacharya-Mesner rank of third order hypermatrices