New Algorithm for Determinant of Matrices Via Combinatorial Approach - Zake, Lugen M. (2012)
Methods for finding determinants for matrices have long been explored and attracted interest of numerous researchers. However, most of the existing methods are tedious and require lengthy computation time particularly as the size of matrices becomes larger. Therefore, this study attempts to develop a new method which can reduce determinant computation time for matrices of any order. The developed method was based on permutations which were generated using starter sets. All starter sets for n elements were obtained by using combinatorial approach which then produced all n! distinct permutations. This starter sets strategy was proven to be more efficient if compared to other existing methods for listing all permutations such as lexicographic method. All permutations obtained were then used to construct a new method for finding determinants of n x n matrices. This study also produced a new theorem for finding determinant of n x n matrices and this theorem was proven to be equivalent to the existing theorem i.e Leibniz theorem. Besides that, several new theoretical works and mathematical properties for generating permutation and determining determinant were also constructed to verify the new developed method. The numerical results revealed that the determinant computation time for the new method was faster if compared to the existing methods. Testing of the new method on several special matrices such as Toeplitz, Hilbert and Hessenberg matrices was also carried out to prove the efficiency of the developed method. The numerical results also indicated that the new method outperformed Gauss and Gauss Jordon methods in term of computation time.