. . . "Ekspansi Laplace"@in . . . . . . . . . . . . . . . . . . . . . . . . "Em \u00E1lgebra linear, o teorema de Laplace fornece uma express\u00E3o para o determinante de uma matriz quadrada qualquer em termos de determinantes de matrizes de ordem inferior."@pt . . "El teorema de Laplace (tamb\u00E9 conegut com a expansi\u00F3 de Laplace o desenvolupament de Laplace ), aix\u00ED anomenat en honor del matem\u00E0tic franc\u00E8s hom\u00F2nim un teorema matem\u00E0tic que permet simplificar el c\u00E0lcul de determinants en matrius d'elevades dimensions per mitj\u00E0 de descompondre'l en la suma de determinants menors. El teorema afirma que el determinant d'una matriu \u00E9s igual a la suma dels determinants dels adjunts de qualsevol fila o columna de la matriu, la qual cosa redueix un determinant de dimensi\u00F3 n en determinants de dimensi\u00F3 n-1. Aplicat de manera successiva, permet arribar a matrius 3x3 (amb el que es pot aplicar la regla de Sarrus) o 2x2 (en el qual el determinant \u00E9s el producte de la diagonal principal menys el de la secund\u00E0ria). Es pot optimitzar els c\u00E0lculs aplicant la i fent zeros el que redueix el nombre de determinants de rang inferior a calcular."@ca . . "\u0422\u0435\u043E\u0440\u0435\u0301\u043C\u0430 \u041B\u0430\u043F\u043B\u0430\u0301\u0441\u0430 \u2014 \u043E\u0434\u043D\u0430 \u0438\u0437 \u0442\u0435\u043E\u0440\u0435\u043C \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u044B. \u041D\u0430\u0437\u0432\u0430\u043D\u0430 \u0432 \u0447\u0435\u0441\u0442\u044C \u0444\u0440\u0430\u043D\u0446\u0443\u0437\u0441\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u041F\u044C\u0435\u0440\u0430-\u0421\u0438\u043C\u043E\u043D\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 (1749 \u2014 1827), \u043A\u043E\u0442\u043E\u0440\u043E\u043C\u0443 \u043F\u0440\u0438\u043F\u0438\u0441\u044B\u0432\u0430\u044E\u0442 \u0444\u043E\u0440\u043C\u0443\u043B\u0438\u0440\u043E\u0432\u0430\u043D\u0438\u0435 \u044D\u0442\u043E\u0439 \u0442\u0435\u043E\u0440\u0435\u043C\u044B \u0432 1772 \u0433\u043E\u0434\u0443, \u0445\u043E\u0442\u044F \u0447\u0430\u0441\u0442\u043D\u044B\u0439 \u0441\u043B\u0443\u0447\u0430\u0439 \u044D\u0442\u043E\u0439 \u0442\u0435\u043E\u0440\u0435\u043C\u044B \u043E \u0440\u0430\u0437\u043B\u043E\u0436\u0435\u043D\u0438\u0438 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0438\u0442\u0435\u043B\u044F \u043F\u043E \u0441\u0442\u0440\u043E\u043A\u0435 (\u0441\u0442\u043E\u043B\u0431\u0446\u0443) \u0431\u044B\u043B \u0438\u0437\u0432\u0435\u0441\u0442\u0435\u043D \u0435\u0449\u0451 \u041B\u0435\u0439\u0431\u043D\u0438\u0446\u0443."@ru . . . . . . . . . "In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n \u00D7 n matrix B as a weighted sum of minors, which are the determinants of some (n \u2212 1) \u00D7 (n \u2212 1) submatrices of B. Specifically, for every i, where is the entry of the ith row and jth column of B, and is the determinant of the submatrix obtained by removing the ith row and the jth column of B. The term is called the cofactor of in B."@en . "Teorema di Laplace"@it . . . . "El teorema de Laplace (tambi\u00E9n conocido como regla de Laplace o desarrollo de Laplace), as\u00ED llamado en honor del matem\u00E1tico franc\u00E9s hom\u00F3nimo es un teorema matem\u00E1tico que permite simplificar el c\u00E1lculo de determinantes en matrices de elevadas dimensiones a base de descomponerlo en la suma de determinantes menores. Se puede optimizar los c\u00E1lculos aplicando la y haciendo ceros lo que reduce el n\u00FAmero de determinantes de rango inferior a calcular."@es . . "\u6570\u5B66\u306E\u7DDA\u578B\u4EE3\u6570\u5B66\u306B\u304A\u3051\u308B\u4F59\u56E0\u5B50\u5C55\u958B\uFF08\u3088\u3044\u3093\u3057\u3066\u3093\u304B\u3044\u3001\u82F1: cofactor expansion\uFF09\u3001\u3042\u308B\u3044\u306F\u30D4\u30A8\u30FC\u30EB\u30FB\u30B7\u30E2\u30F3\u30FB\u30E9\u30D7\u30E9\u30B9\u306E\u540D\u306B\u56E0\u3093\u3067\u30E9\u30D7\u30E9\u30B9\u5C55\u958B\u3068\u306F\u3001n\u6B21\u6B63\u65B9\u884C\u5217 A \u306E\u884C\u5217\u5F0F |A| \u306E\u3001n \u500B\u306E A \u306E (n \u2212 1)\u6B21\u5C0F\u884C\u5217\u5F0F\u306E\u91CD\u307F\u4ED8\u304D\u548C\u3068\u3057\u3066\u306E\u8868\u793A\u3067\u3042\u308B\u3002\u4F59\u56E0\u5B50\u5C55\u958B\u306F\u884C\u5217\u5F0F\u3092\u898B\u308B\u3044\u304F\u3064\u304B\u306E\u65B9\u6CD5\u306E\u4E00\u3064\u3068\u3057\u3066\u7406\u8AD6\u7684\u306B\u8208\u5473\u6DF1\u304F\u3001\u884C\u5217\u5F0F\u306E\u5B9F\u969B\u306E\u8A08\u7B97\u306B\u304A\u3044\u3066\u3082\u6709\u7528\u3067\u3042\u308B\u3002 A \u306E (i, j)\u4F59\u56E0\u5B50\u3068\u306F\u3001\u6B21\u3067\u5B9A\u7FA9\u3055\u308C\u308B\u30B9\u30AB\u30E9\u30FC\u3067\u3042\u308B\uFF1A \u3053\u3053\u3067 Mi,j \u306F A \u306E (i, j)\u5C0F\u884C\u5217\u5F0F\u3001\u3064\u307E\u308A\u3001A \u304B\u3089\u7B2Ci\u884C\u3068\u7B2Cj\u5217\u3092\u9664\u3044\u3066\u5F97\u3089\u308C\u308B (n \u2212 1)\u6B21\u5C0F\u6B63\u65B9\u884C\u5217\u306E\u884C\u5217\u5F0F\u3067\u3042\u308B\u3002 \u3059\u308B\u3068\u4F59\u56E0\u5B50\u5C55\u958B\u306F\u6B21\u3067\u4E0E\u3048\u3089\u308C\u308B\uFF1A \u5B9A\u7406 \u2015 A = (ai,j) \u3092 n\u6B21\u6B63\u65B9\u884C\u5217\u3068\u3057\u3001\u4EFB\u610F\u306E i, j \u2208 {1, 2, \u2026, n} \u3092\u56FA\u5B9A\u3059\u308B\u3002 \u3059\u308B\u3068\u305D\u306E\u884C\u5217\u5F0F |A| \u306F\u6B21\u3067\u4E0E\u3048\u3089\u308C\u308B\uFF1A"@ja . "Em \u00E1lgebra linear, o teorema de Laplace fornece uma express\u00E3o para o determinante de uma matriz quadrada qualquer em termos de determinantes de matrizes de ordem inferior."@pt . "Dalam aljabar linear, ekspansi Laplace, dinamai Pierre-Simon Laplace, juga disebut ekspansi kofaktor, adalah ekspresi dari determinan n \u00D7 n matriks B sebagai jumlah tertimbang dari minor, yang merupakan determinan dari beberapa B submatriks B. Secara khusus, untuk setiap i, dimana adalah entri baris ke-i dan kolom ke-j dari B, dan adalah determinan submatriks yang diperoleh dengan menghilangkan baris ke-i dan kolom ke-j dari B. Syarat disebut kofaktor dari di B."@in . . . "In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n \u00D7 n matrix B as a weighted sum of minors, which are the determinants of some (n \u2212 1) \u00D7 (n \u2212 1) submatrices of B. Specifically, for every i, where is the entry of the ith row and jth column of B, and is the determinant of the submatrix obtained by removing the ith row and the jth column of B. The term is called the cofactor of in B. The Laplace expansion is often useful in proofs, as in, for example, allowing recursion on the size of matrices. It is also of didactic interest for its simplicity, and as one of several ways to view and compute the determinant. For large matrices, it quickly becomes inefficient to compute, when compared to Gaussian elimination."@en . . "12516"^^ . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430"@uk . "In matematica, in particolare in algebra lineare, il teorema di Laplace o sviluppo di Laplace, il cui nome \u00E8 dovuto a Pierre Simon Laplace, \u00E8 una formula che permette di calcolare il determinante di una matrice (quadrata) con un procedimento ricorsivo. Lo sviluppo pu\u00F2 essere eseguito per righe oppure per colonne."@it . "In matematica, in particolare in algebra lineare, il teorema di Laplace o sviluppo di Laplace, il cui nome \u00E8 dovuto a Pierre Simon Laplace, \u00E8 una formula che permette di calcolare il determinante di una matrice (quadrata) con un procedimento ricorsivo. Lo sviluppo pu\u00F2 essere eseguito per righe oppure per colonne."@it . . . . . "\u5728\u6570\u5B66\u4E2D\uFF0C\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\uFF08\u6216\u79F0\u62C9\u666E\u62C9\u65AF\u516C\u5F0F\uFF09\u662F\u4E00\u4E2A\u5173\u4E8E\u884C\u5217\u5F0F\u7684\u5C55\u5F00\u5F0F\u3002\u5C06\u4E00\u4E2An\u00D7n\u77E9\u9635B\u7684\u884C\u5217\u5F0F\u8FDB\u884C\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\uFF0C\u5373\u662F\u5C06\u5176\u8868\u793A\u6210\u5173\u4E8E\u77E9\u9635B\u7684\u67D0\u4E00\u884C\uFF08\u6216\u67D0\u4E00\u5217\uFF09\u7684n\u4E2A\u5143\u7D20\u7684(n-1)\u00D7(n-1)\u4F59\u5B50\u5F0F\u7684\u548C\u3002\u884C\u5217\u5F0F\u7684\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\u4E00\u822C\u88AB\u7B80\u79F0\u4E3A\u884C\u5217\u5F0F\u6309\u67D0\u4E00\u884C\uFF08\u6216\u6309\u67D0\u4E00\u5217\uFF09\u7684\u5C55\u5F00\u3002\u7531\u4E8E\u77E9\u9635B\u6709n\u884Cn\u5217\uFF0C\u5B83\u7684\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\u4E00\u5171\u67092n\u79CD\u3002\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\u7684\u63A8\u5E7F\u79F0\u4E3A\u62C9\u666E\u62C9\u65AF\u5B9A\u7406\uFF0C\u662F\u5C06\u4E00\u884C\u7684\u5143\u7D20\u63A8\u5E7F\u4E3A\u5173\u4E8Ek\u884C\u7684\u4E00\u5207\u5B50\u5F0F\u3002\u5B83\u4EEC\u7684\u6BCF\u4E00\u9879\u548C\u5BF9\u5E94\u7684\u4EE3\u6570\u4F59\u5B50\u5F0F\u7684\u4E58\u79EF\u4E4B\u548C\u4ECD\u7136\u662FB\u7684\u884C\u5217\u5F0F\u3002\u7814\u7A76\u4E00\u4E9B\u7279\u5B9A\u7684\u5C55\u5F00\u53EF\u4EE5\u51CF\u5C11\u5BF9\u4E8E\u77E9\u9635B\u4E4B\u884C\u5217\u5F0F\u7684\u8BA1\u7B97\uFF0C\u62C9\u666E\u62C9\u65AF\u516C\u5F0F\u4E5F\u5E38\u7528\u4E8E\u4E00\u4E9B\u62BD\u8C61\u7684\u63A8\u5BFC\u4E2D\u3002"@zh . . . . . . "Teorema de Laplace"@pt . "Teorema de Laplace"@es . . . . "\uC120\uD615\uB300\uC218\uD559\uC5D0\uC11C \uB77C\uD50C\uB77C\uC2A4 \uC804\uAC1C(-\u5C55\u958B, \uC601\uC5B4: Laplace expansion) \uB610\uB294 \uC5EC\uC778\uC790 \uC804\uAC1C(\u9918\u56E0\u5B50\u5C55\u958B, \uC601\uC5B4: cofactor expansion)\uB294 \uD589\uB82C\uC2DD\uC744 \uB354 \uC791\uC740 \uB450 \uD589\uB82C\uC2DD\uACFC \uADF8\uC5D0 \uB9DE\uB294 \uBD80\uD638\uB97C \uACF1\uD55C \uAC83\uB4E4\uC758 \uD569\uC73C\uB85C \uC804\uAC1C\uD558\uB294 \uAC83\uC774\uB2E4."@ko . "El teorema de Laplace (tambi\u00E9n conocido como regla de Laplace o desarrollo de Laplace), as\u00ED llamado en honor del matem\u00E1tico franc\u00E9s hom\u00F3nimo es un teorema matem\u00E1tico que permite simplificar el c\u00E1lculo de determinantes en matrices de elevadas dimensiones a base de descomponerlo en la suma de determinantes menores. El teorema afirma que el determinante de una matriz es igual a la suma de los productos de cada elemento (de un rengl\u00F3n o columna) por la determinante de su matriz adjunta, lo que reduce un determinante de dimensi\u00F3n n a n determinantes de dimensi\u00F3n n-1. Aplicado de forma sucesiva, permite llegar a matrices 3x3 (con lo que se puede aplicar la regla de Sarrus) o 2x2 (en el que el determinante es el producto de la diagonal principal menos el de la secundaria). Se puede optimizar los c\u00E1lculos aplicando la y haciendo ceros lo que reduce el n\u00FAmero de determinantes de rango inferior a calcular."@es . . . . . "Laplace expansion"@en . . . . . "\u0422\u0435\u043E\u0440\u0435\u0301\u043C\u0430 \u041B\u0430\u043F\u043B\u0430\u0301\u0441\u0430 (\u0440\u043E\u0437\u043A\u043B\u0430\u0434 \u041B\u0430\u043F\u043B\u0430\u0441\u0430) \u2014 \u043E\u0434\u043D\u0430 \u0437 \u0442\u0435\u043E\u0440\u0435\u043C \u0432 \u0442\u0435\u043E\u0440\u0456\u0457 \u043C\u0430\u0442\u0440\u0438\u0446\u044C. \u041D\u0430\u0437\u0432\u0430\u043D\u0430 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0444\u0440\u0430\u043D\u0446\u0443\u0437\u044C\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u041F'\u0454\u0440\u0430-\u0421\u0438\u043C\u043E\u043D\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430, \u044F\u043A\u043E\u043C\u0443 \u043F\u0440\u0438\u043F\u0438\u0441\u0443\u044E\u0442\u044C \u0434\u043E\u0432\u0435\u0434\u0435\u043D\u043D\u044F \u0446\u0456\u0454\u0457 \u0442\u0435\u043E\u0440\u0435\u043C\u0438 \u0432 1772 \u0440\u043E\u0446\u0456, \u0445\u043E\u0447\u0430 \u043E\u043A\u0440\u0435\u043C\u0438\u0439 \u0432\u0438\u043F\u0430\u0434\u043E\u043A \u0446\u0456\u0454\u0457 \u0442\u0435\u043E\u0440\u0435\u043C\u0438 \u043F\u0440\u043E \u0440\u043E\u0437\u043A\u043B\u0430\u0434\u0430\u043D\u043D\u044F \u0432\u0438\u0437\u043D\u0430\u0447\u043D\u0438\u043A\u0430 \u043F\u043E \u0440\u044F\u0434\u043A\u0443 (\u0441\u0442\u043E\u0432\u043F\u0446\u044E) \u0431\u0443\u0432 \u0432\u0456\u0434\u043E\u043C\u0438\u0439 \u0449\u0435 \u041B\u0435\u0439\u0431\u043D\u0456\u0446\u0443."@uk . . "\uB77C\uD50C\uB77C\uC2A4 \uC804\uAC1C"@ko . . . "El teorema de Laplace (tamb\u00E9 conegut com a expansi\u00F3 de Laplace o desenvolupament de Laplace ), aix\u00ED anomenat en honor del matem\u00E0tic franc\u00E8s hom\u00F2nim un teorema matem\u00E0tic que permet simplificar el c\u00E0lcul de determinants en matrius d'elevades dimensions per mitj\u00E0 de descompondre'l en la suma de determinants menors. Es pot optimitzar els c\u00E0lculs aplicant la i fent zeros el que redueix el nombre de determinants de rang inferior a calcular."@ca . . "\uC120\uD615\uB300\uC218\uD559\uC5D0\uC11C \uB77C\uD50C\uB77C\uC2A4 \uC804\uAC1C(-\u5C55\u958B, \uC601\uC5B4: Laplace expansion) \uB610\uB294 \uC5EC\uC778\uC790 \uC804\uAC1C(\u9918\u56E0\u5B50\u5C55\u958B, \uC601\uC5B4: cofactor expansion)\uB294 \uD589\uB82C\uC2DD\uC744 \uB354 \uC791\uC740 \uB450 \uD589\uB82C\uC2DD\uACFC \uADF8\uC5D0 \uB9DE\uB294 \uBD80\uD638\uB97C \uACF1\uD55C \uAC83\uB4E4\uC758 \uD569\uC73C\uB85C \uC804\uAC1C\uD558\uB294 \uAC83\uC774\uB2E4."@ko . "\u5728\u6570\u5B66\u4E2D\uFF0C\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\uFF08\u6216\u79F0\u62C9\u666E\u62C9\u65AF\u516C\u5F0F\uFF09\u662F\u4E00\u4E2A\u5173\u4E8E\u884C\u5217\u5F0F\u7684\u5C55\u5F00\u5F0F\u3002\u5C06\u4E00\u4E2An\u00D7n\u77E9\u9635B\u7684\u884C\u5217\u5F0F\u8FDB\u884C\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\uFF0C\u5373\u662F\u5C06\u5176\u8868\u793A\u6210\u5173\u4E8E\u77E9\u9635B\u7684\u67D0\u4E00\u884C\uFF08\u6216\u67D0\u4E00\u5217\uFF09\u7684n\u4E2A\u5143\u7D20\u7684(n-1)\u00D7(n-1)\u4F59\u5B50\u5F0F\u7684\u548C\u3002\u884C\u5217\u5F0F\u7684\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\u4E00\u822C\u88AB\u7B80\u79F0\u4E3A\u884C\u5217\u5F0F\u6309\u67D0\u4E00\u884C\uFF08\u6216\u6309\u67D0\u4E00\u5217\uFF09\u7684\u5C55\u5F00\u3002\u7531\u4E8E\u77E9\u9635B\u6709n\u884Cn\u5217\uFF0C\u5B83\u7684\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\u4E00\u5171\u67092n\u79CD\u3002\u62C9\u666E\u62C9\u65AF\u5C55\u5F00\u7684\u63A8\u5E7F\u79F0\u4E3A\u62C9\u666E\u62C9\u65AF\u5B9A\u7406\uFF0C\u662F\u5C06\u4E00\u884C\u7684\u5143\u7D20\u63A8\u5E7F\u4E3A\u5173\u4E8Ek\u884C\u7684\u4E00\u5207\u5B50\u5F0F\u3002\u5B83\u4EEC\u7684\u6BCF\u4E00\u9879\u548C\u5BF9\u5E94\u7684\u4EE3\u6570\u4F59\u5B50\u5F0F\u7684\u4E58\u79EF\u4E4B\u548C\u4ECD\u7136\u662FB\u7684\u884C\u5217\u5F0F\u3002\u7814\u7A76\u4E00\u4E9B\u7279\u5B9A\u7684\u5C55\u5F00\u53EF\u4EE5\u51CF\u5C11\u5BF9\u4E8E\u77E9\u9635B\u4E4B\u884C\u5217\u5F0F\u7684\u8BA1\u7B97\uFF0C\u62C9\u666E\u62C9\u65AF\u516C\u5F0F\u4E5F\u5E38\u7528\u4E8E\u4E00\u4E9B\u62BD\u8C61\u7684\u63A8\u5BFC\u4E2D\u3002"@zh . . . "Rozwini\u0119cie Laplace\u2019a"@pl . . . . . "Rozwini\u0119cie Laplace\u2019a \u2013 wz\u00F3r rekurencyjny okre\u015Blaj\u0105cy wyznacznik -tego stopnia macierzy kwadratowej o wymiarach Nazwa wzoru pochodzi od francuskiego matematyka Laplace\u2019a. Niech W\u00F3wczas: \n* dla ka\u017Cdego ustalonego zachodzi \n* dla ka\u017Cdego ustalonego zachodzi gdzie: jest elementem macierzy w -tym wierszu i -tej kolumnie, jest dope\u0142nieniem algebraicznym elementu Powy\u017Csze wzory nazywamy rozwini\u0119ciami Laplace\u2019a, pierwszy wzgl\u0119dem -tej kolumny, a drugi wzgl\u0119dem -tego wiersza."@pl . . "3506553"^^ . "\u0422\u0435\u043E\u0440\u0435\u0301\u043C\u0430 \u041B\u0430\u043F\u043B\u0430\u0301\u0441\u0430 (\u0440\u043E\u0437\u043A\u043B\u0430\u0434 \u041B\u0430\u043F\u043B\u0430\u0441\u0430) \u2014 \u043E\u0434\u043D\u0430 \u0437 \u0442\u0435\u043E\u0440\u0435\u043C \u0432 \u0442\u0435\u043E\u0440\u0456\u0457 \u043C\u0430\u0442\u0440\u0438\u0446\u044C. \u041D\u0430\u0437\u0432\u0430\u043D\u0430 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0444\u0440\u0430\u043D\u0446\u0443\u0437\u044C\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u041F'\u0454\u0440\u0430-\u0421\u0438\u043C\u043E\u043D\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430, \u044F\u043A\u043E\u043C\u0443 \u043F\u0440\u0438\u043F\u0438\u0441\u0443\u044E\u0442\u044C \u0434\u043E\u0432\u0435\u0434\u0435\u043D\u043D\u044F \u0446\u0456\u0454\u0457 \u0442\u0435\u043E\u0440\u0435\u043C\u0438 \u0432 1772 \u0440\u043E\u0446\u0456, \u0445\u043E\u0447\u0430 \u043E\u043A\u0440\u0435\u043C\u0438\u0439 \u0432\u0438\u043F\u0430\u0434\u043E\u043A \u0446\u0456\u0454\u0457 \u0442\u0435\u043E\u0440\u0435\u043C\u0438 \u043F\u0440\u043E \u0440\u043E\u0437\u043A\u043B\u0430\u0434\u0430\u043D\u043D\u044F \u0432\u0438\u0437\u043D\u0430\u0447\u043D\u0438\u043A\u0430 \u043F\u043E \u0440\u044F\u0434\u043A\u0443 (\u0441\u0442\u043E\u0432\u043F\u0446\u044E) \u0431\u0443\u0432 \u0432\u0456\u0434\u043E\u043C\u0438\u0439 \u0449\u0435 \u041B\u0435\u0439\u0431\u043D\u0456\u0446\u0443."@uk . "\u0422\u0435\u043E\u0440\u0435\u0301\u043C\u0430 \u041B\u0430\u043F\u043B\u0430\u0301\u0441\u0430 \u2014 \u043E\u0434\u043D\u0430 \u0438\u0437 \u0442\u0435\u043E\u0440\u0435\u043C \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u044B. \u041D\u0430\u0437\u0432\u0430\u043D\u0430 \u0432 \u0447\u0435\u0441\u0442\u044C \u0444\u0440\u0430\u043D\u0446\u0443\u0437\u0441\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u041F\u044C\u0435\u0440\u0430-\u0421\u0438\u043C\u043E\u043D\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430 (1749 \u2014 1827), \u043A\u043E\u0442\u043E\u0440\u043E\u043C\u0443 \u043F\u0440\u0438\u043F\u0438\u0441\u044B\u0432\u0430\u044E\u0442 \u0444\u043E\u0440\u043C\u0443\u043B\u0438\u0440\u043E\u0432\u0430\u043D\u0438\u0435 \u044D\u0442\u043E\u0439 \u0442\u0435\u043E\u0440\u0435\u043C\u044B \u0432 1772 \u0433\u043E\u0434\u0443, \u0445\u043E\u0442\u044F \u0447\u0430\u0441\u0442\u043D\u044B\u0439 \u0441\u043B\u0443\u0447\u0430\u0439 \u044D\u0442\u043E\u0439 \u0442\u0435\u043E\u0440\u0435\u043C\u044B \u043E \u0440\u0430\u0437\u043B\u043E\u0436\u0435\u043D\u0438\u0438 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0438\u0442\u0435\u043B\u044F \u043F\u043E \u0441\u0442\u0440\u043E\u043A\u0435 (\u0441\u0442\u043E\u043B\u0431\u0446\u0443) \u0431\u044B\u043B \u0438\u0437\u0432\u0435\u0441\u0442\u0435\u043D \u0435\u0449\u0451 \u041B\u0435\u0439\u0431\u043D\u0438\u0446\u0443."@ru . . "\u6570\u5B66\u306E\u7DDA\u578B\u4EE3\u6570\u5B66\u306B\u304A\u3051\u308B\u4F59\u56E0\u5B50\u5C55\u958B\uFF08\u3088\u3044\u3093\u3057\u3066\u3093\u304B\u3044\u3001\u82F1: cofactor expansion\uFF09\u3001\u3042\u308B\u3044\u306F\u30D4\u30A8\u30FC\u30EB\u30FB\u30B7\u30E2\u30F3\u30FB\u30E9\u30D7\u30E9\u30B9\u306E\u540D\u306B\u56E0\u3093\u3067\u30E9\u30D7\u30E9\u30B9\u5C55\u958B\u3068\u306F\u3001n\u6B21\u6B63\u65B9\u884C\u5217 A \u306E\u884C\u5217\u5F0F |A| \u306E\u3001n \u500B\u306E A \u306E (n \u2212 1)\u6B21\u5C0F\u884C\u5217\u5F0F\u306E\u91CD\u307F\u4ED8\u304D\u548C\u3068\u3057\u3066\u306E\u8868\u793A\u3067\u3042\u308B\u3002\u4F59\u56E0\u5B50\u5C55\u958B\u306F\u884C\u5217\u5F0F\u3092\u898B\u308B\u3044\u304F\u3064\u304B\u306E\u65B9\u6CD5\u306E\u4E00\u3064\u3068\u3057\u3066\u7406\u8AD6\u7684\u306B\u8208\u5473\u6DF1\u304F\u3001\u884C\u5217\u5F0F\u306E\u5B9F\u969B\u306E\u8A08\u7B97\u306B\u304A\u3044\u3066\u3082\u6709\u7528\u3067\u3042\u308B\u3002 A \u306E (i, j)\u4F59\u56E0\u5B50\u3068\u306F\u3001\u6B21\u3067\u5B9A\u7FA9\u3055\u308C\u308B\u30B9\u30AB\u30E9\u30FC\u3067\u3042\u308B\uFF1A \u3053\u3053\u3067 Mi,j \u306F A \u306E (i, j)\u5C0F\u884C\u5217\u5F0F\u3001\u3064\u307E\u308A\u3001A \u304B\u3089\u7B2Ci\u884C\u3068\u7B2Cj\u5217\u3092\u9664\u3044\u3066\u5F97\u3089\u308C\u308B (n \u2212 1)\u6B21\u5C0F\u6B63\u65B9\u884C\u5217\u306E\u884C\u5217\u5F0F\u3067\u3042\u308B\u3002 \u3059\u308B\u3068\u4F59\u56E0\u5B50\u5C55\u958B\u306F\u6B21\u3067\u4E0E\u3048\u3089\u308C\u308B\uFF1A \u5B9A\u7406 \u2015 A = (ai,j) \u3092 n\u6B21\u6B63\u65B9\u884C\u5217\u3068\u3057\u3001\u4EFB\u610F\u306E i, j \u2208 {1, 2, \u2026, n} \u3092\u56FA\u5B9A\u3059\u308B\u3002 \u3059\u308B\u3068\u305D\u306E\u884C\u5217\u5F0F |A| \u306F\u6B21\u3067\u4E0E\u3048\u3089\u308C\u308B\uFF1A"@ja . . "Dalam aljabar linear, ekspansi Laplace, dinamai Pierre-Simon Laplace, juga disebut ekspansi kofaktor, adalah ekspresi dari determinan n \u00D7 n matriks B sebagai jumlah tertimbang dari minor, yang merupakan determinan dari beberapa B submatriks B. Secara khusus, untuk setiap i, dimana adalah entri baris ke-i dan kolom ke-j dari B, dan adalah determinan submatriks yang diperoleh dengan menghilangkan baris ke-i dan kolom ke-j dari B. Syarat disebut kofaktor dari di B."@in . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u041B\u0430\u043F\u043B\u0430\u0441\u0430"@ru . "1098519075"^^ . . "\u62C9\u666E\u62C9\u65AF\u5C55\u5F00"@zh . "\u4F59\u56E0\u5B50\u5C55\u958B"@ja . . . . "Rozwini\u0119cie Laplace\u2019a \u2013 wz\u00F3r rekurencyjny okre\u015Blaj\u0105cy wyznacznik -tego stopnia macierzy kwadratowej o wymiarach Nazwa wzoru pochodzi od francuskiego matematyka Laplace\u2019a. Niech W\u00F3wczas: \n* dla ka\u017Cdego ustalonego zachodzi \n* dla ka\u017Cdego ustalonego zachodzi gdzie: jest elementem macierzy w -tym wierszu i -tej kolumnie, jest dope\u0142nieniem algebraicznym elementu Powy\u017Csze wzory nazywamy rozwini\u0119ciami Laplace\u2019a, pierwszy wzgl\u0119dem -tej kolumny, a drugi wzgl\u0119dem -tego wiersza."@pl . . "Teorema de Laplace"@ca . . . .