The applications of the bideterminant of a matrix over commutative semirings
Qianyu Shu, Xue-ping Wang · Linear and Multilinear Algebra · 2016
This paper discusses some applications of the bideterminant of a matrix over commutative semirings. It first uses the bideterminant of a matrix to define the McCoy rank of a matrix and gives some necessary and sufficient conditions that the McCoy rank of an matrix is less than n. It then uses those conditions to investigate the relationships among linear dependence of vectors, linear independence of vectors and the bideterminant of a matrix. It finally shows that the generalized Cramer’s rule is valid for solving the system of linear equations.