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Area Mathematics - General concepts and linear algebra / Matrices

IEV ref102-06-20

en
determinant, <of a matrix>
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@ .


fr
déterminant, <d'une matrice> m
pour une matrice carrée A d'ordre n et d'éléments Aij, scalaire noté det A, égal à la somme algébrique des produits obtenus en prenant comme facteurs de toutes les manières possibles un élément et un seul dans chaque ligne et dans chaque colonne, chacun de ces produits étant affectés du signe plus ou du signe moins suivant que le nombre total des inversions des deux indices est pair ou impair

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@

σ = (σ(1), σ(2), …, σ(n)) est une permutation des indices (1, 2, …, n), ε(σ) est le nombre d’inversions dans la permutation σ, et la somme notée Σ est étendue à toutes les permutations

Note 1 à l'article: Le déterminant d'une matrice est égal au déterminant des n vecteurs dont les coordonnées sont les éléments des lignes ou des colonnes.

Note 2 à l'article: Le déterminant de la matrice 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@ est noté det A ou | 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@ .


de
Determinante (einer Matrix), f

es
determinante (de una matriz)

ja
行列式, <行列の>

pl
wyznacznik (macierzy)

pt
determinante (de uma matriz)

sr
детерминанта, <матрице> ж јд

sv
determinant (av en matris)

zh
行列式, <矩阵的>

Publication date: 2008-08
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