Number of Non-Unique Minors (of Various Orders) and Elements in the Calculation of General Determinants
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Problem statement: Many distinct properties of determinants have been studied and are known, yet a considerable number of properties still need further examination. This study investigates the number of minors (of various orders) and elements of a matrix A contained in the expansion of the general determinant of A, irrespective of the independence, principality and distinctness of such minors and elements. Approach: A mathematical proof based approach is taken. Minors of all orders and elements in the calculations of general determinants of matrices of sizes 2䲬 3䳬 4䴍 and 5䵠respectively, are considered. Results: Two general expressions involving factorial terms are found: the first being equivalent to the number of minors of various orders found in the analysis of the considered matrices (mentioned above) and the second being equivalent to the number of elements found in the same analysis. Proofs are then presented showing that the expressions hold in the general case of a matrix of size n䮮 Conclusion: The results of this study present, with proof, expressions for the total number of minors (of various orders) and elements, respectively, in the general determinant of a matrix of size n䮬 irrespective of the independence, principality and distinctness of such minors and elements. Scope for further theoretical study, with applications in applied mathematics and the physical and computer sciences is also indicated.
Journal of Mathematics and Statistics
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Algebra and Number Theory
Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)