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IEVref:102-06-20ID:
Language:enStatus: Standard
Term: determinant, <of a matrix>
Synonym1:
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Definition: for a square matrix A of order n with the elements Aij, scalar denoted by det A, equal to the algebraic sum of the products obtained by taking as factors in all possible ways one and only one element from each row and each column, each product with the sign plus or minus depending whether the total number of inversions of the two subscripts is even or odd

detA= σ (1) ε(σ) A 1σ(1) A 2σ(2) ... A nσ(n) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaaiizaiaacw gacaGG0bGaaGjbVRGaaGPaVlaahgeacqGH9aqpdaaeqbqaaiaacIca cqGHsisljugqbiaaigdakiaacMcadaahaaWcbeqaaiabew7aLjaays W7caGGOaGaeq4WdmNaaiykaaaakiaadgeadaWgaaWcbaqcLboacaaI XaWccaaMc8Uaeq4WdmNaaGPaVlaacIcacaaIXaGaaiykaaqabaaaba Gaeq4WdmhabeqdcqGHris5aOGaamyqamaaBaaaleaajug4aiaaikda liaaykW7cqaHdpWCcaaMc8UaaiikaiaaikdacaGGPaaabeaakiaac6 cacaGGUaGaaiOlaiaadgeadaWgaaWcbaGaamOBaiaaykW7cqaHdpWC caaMe8Uaaiikaiaad6gacaGGPaaabeaaaaa@6E71@

where σ = (σ(1), σ(2), …, σ(n)) is a permutation of the subscripts (1, 2, …, n), ε(σ) is the number of inversions in permutation σ, and the sum denoted by Σ is for all permutations

Note 1 to entry: The determinant of a matrix is equal to the determinant of the n vectors, the coordinates of which are the elements of the rows or of the columns.

Note 2 to entry: The determinant of the matrix A=( A 11 A 1n A n1 A nn ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahgeacqGH9aqpda qadaqaauaabeqadmaaaeaacaWGbbWaaSbaaSqaaKqzGdGaaGymaiaa igdaaSqabaaakeaacqWIVlctaeaacaWGbbWaaSbaaSqaaKqzGdGaaG ymaSGaamOBaaqabaaakeaacqWIUlstaeaacqWIXlYtaeaacqWIUlst aeaacaWGbbWaaSbaaSqaaiaad6gajug4aiaaigdaaSqabaaakeaacq WIVlctaeaacaWGbbWaaSbaaSqaaiaad6gacaWGUbaabeaaaaaakiaa wIcacaGLPaaaaaa@549B@ is denoted A or | A 11 A 1n A n1 A nn | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaaysW7caaMc8+aaq WaaeaafaqabeWadaaabaGaamyqamaaBaaaleaajug4aiaaigdacaaI XaaaleqaaaGcbaGaeS47IWeabaGaamyqamaaBaaaleaajug4aiaaig daliaad6gaaeqaaaGcbaGaeSO7I0eabaGaeSy8I8eabaGaeSO7I0ea baGaamyqamaaBaaaleaacaWGUbqcLboacaaIXaaaleqaaaGcbaGaeS 47IWeabaGaamyqamaaBaaaleaacaWGUbGaamOBaaqabaaaaaGccaGL hWUaayjcSdaaaa@577C@ .


Publication date:2008-08
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Internal notes:2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO
2017-08-25: Corrected order of i and sub tags. LMO
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