Document Type : Original Manuscript


Department of Mathematics, K M M Government Women’s College, Kannur, P.O. Box 670004, Kerala, India


Given a matrix A with elements from a field of characteristic zero, the transin-
verse A# of A is defined as the transpose of the matrix obtained by replacing the
non-zero elements of A by their inverses and leaving zeros, if any, unchanged.
We discuss the properties of this matrix operation in some detail and as an important application, we reinvent the celebrated matrix tree theorem for gain graphs.
Characterization of balance in connected gain graphs using its Laplacian matrix becomes an immediate consequence.


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