Shahrood University of TechnologyJournal of Algebraic Systems2345-512812120240901ON TRANSINVERSE OF MATRICES AND ITS APPLICATIONS135147284210.22044/jas.2022.12107.1629ENKoombail Shahul HameedDepartment of Mathematics, K M M Government Women’s College, Kannur, P.O.
Box 670004, Kerala, India0000-0003-0905-3865Kunhumbidukka Othayoth RamakrishnanDepartment of Mathematics, K M M Government Women’s College, Kannur, P.O.
Box 670004, Kerala, IndiaJournal Article20220713Given a matrix A with elements from a field of characteristic zero, the transin-<br />verse A# of A is defined as the transpose of the matrix obtained by replacing the <br />non-zero elements of A by their inverses and leaving zeros, if any, unchanged.<br />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.<br />Characterization of balance in connected gain graphs using its Laplacian matrix becomes an immediate consequence.https://jas.shahroodut.ac.ir/article_2842_63f0c500dfc9cf0ede99189cacc5ba3a.pdf