Determinant Of Rectangular Matrices And It’s Applications In Graph Theory

Matrix and determinant are very important in modern mathematical theory, and widely applied to other subjects. For matrix, we can learn about the square and non-square matrices, but in Advanced Algebra the calculation of determinant only applies for square matrices. We did not give a definition of the non-square matrix’s determinant. This subject was based on many foreign literature. The foreign literature had given its related definition of non-square matrices. This paper aims to study the determinant of non-square matrices and its application in graph theory and so on.For rectangular matrix’s determinant,we are not very familiar with it. Firstly,this article introduces the definition of rectangular matrix’s determinant and its basic properties. We can get many qualities about rectangular matrices’ determinant and related application. Some of the conclusions are drawn in this paper and the theorem will be proved. For non-square matrix, the paper mainly studies these two types of rectangular matrix’s determinant,which size are n?(n ?1) and n?(n ?2),and researches its application in graph theory. The first class is n rows and n ?1columns rectangular matrices. we can add one row to the rectangular matrix.Then the rectangular matrix can transform into the square matrix and it can be calculated. The other is n rows and n ?2 columns rectangular matrix, and we can add two rows to the matrix and calculate. We also can get the relevant properties.For these two types of rectangular matrix’ determinant, we can research its application in graph theory. First, we can applied the first class of rectangular matrices to the directed tree. we then give weight to the root of tree. And write its associated matrix. We add some appropriate rows to the tree’s matrix so that it can be calculated. Then we get the value and the calculation of such rectangular matrices of directed tree. The value shows the trends of directed tree,the number of layers, the number of vertexes and so on. Starting from the root vertex of directed tree, we carried out the order of reference, and gave weight to the vertexes, and we could directly write the determinant that was based on the certificate of theorems and properties of the directed tree and explain their significance; For the second class rectangular matrices of graph, we add two columns or two rows to the matrices so as it can be calculated. This kind of adding rows to the rectangular matrices is different from the first class. The specific operation is adding two rows or two columns to the rectangular matrices. And it must be n+1 times of cycle and complete the calculations. However, for the second kind of directed tree or directed cycle, we can write the nature and the meaning of its determinant, but because of its proven process is quite complicated, here we only illustrate and apply, and give its properties and the applications of basic directed circle.