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行列式の基本性質について A Remark on the Essential Properties of Determinants 伊藤, 善彦 行列式の基本性質 Let K be a commutative ring. For each n×n matrix A=(aij) over K, we define the determinant d(A) by the formula d(A)=Σσ ε(σ)a(σ), where the sum is taken over the set Sn of all permutations σ of {1,2,…,n}, ε(σ) is the sign of the permutation, and a(σ)=a1σ(1)a2σ(2)…anσ(n) Assume n≧2, and let τ be a transposition in Sn. The mapping γτ: Sn→Sn by the formula γτ(σ)=στ for each σ∈Sn is a bijection. Then it is clear that the following theorem holds : Theorem. d(A) = Σσ (a(σ)-a(στ)), where the sum is taken over the set An of all even permutations σ. In this paper, we will show that next two essential theorems B2 and C1 can be derived from our theorem. Theorem B2. Interchanging two rows of A multiplies d(A) by -1. Theorem C1. If A has two rows alike, then d(A)=0. departmental bulletin paper 浜松医科大学 1987-03-31 VoR application/pdf 浜松医科大学紀要. 一般教育 1 19 22 Reports of Liberal Arts, Hamamatsu University School of Medicine 0914-0174 AN10032827 https://hama-med.repo.nii.ac.jp/record/146/files/kiyo01_02.pdf http://hdl.handle.net/10271/174 https://hama-med.repo.nii.ac.jp/records/146 jpn open access