{"id":10534,"date":"2025-07-15T13:34:49","date_gmt":"2025-07-15T13:34:49","guid":{"rendered":"https:\/\/functiontranslator.com\/macierz-iloczyn\/"},"modified":"2021-02-26T05:21:19","modified_gmt":"2021-02-26T05:21:19","slug":"mmult","status":"publish","type":"post","link":"https:\/\/functiontranslator.com\/pl\/mmult\/","title":{"rendered":"MACIERZ.ILOCZYN"},"content":{"rendered":"

Wprowadzenie do funkcji MACIERZ.ILOCZYN<\/h2>\n

Funkcja MMULT<\/code> (w polskiej wersji Excela i Google Sheets znana jako MACIERZ.ILOCZYN<\/code>) umo\u017cliwia mno\u017cenie dw\u00f3ch macierzy. Jest nieoceniona w wielu dziedzinach, takich jak analiza danych, geometria, ekonomia czy statystyka, gdzie operacje na macierzach s\u0105 stosowane.<\/p>\n

Syntaktyczne podstawy<\/h2>\n

Syntaktyka funkcji MACIERZ.ILOCZYN<\/code> w Microsoft Excel oraz Google Sheets przedstawia si\u0119 nast\u0119puj\u0105co:<\/p>\n

=MACIERZ.ILOCZYN(Macierz1, Macierz2)<\/code><\/pre>\n
    \n
  • Macierz1<\/strong> \u2013 pierwsza macierz do pomno\u017cenia.<\/li>\n
  • Macierz2<\/strong> \u2013 druga macierz do pomno\u017cenia.<\/li>\n<\/ul>\n
    Wymagane jest, aby liczba kolumn w Macierz1<\/code> odpowiada\u0142a liczbie wierszy w Macierz2<\/code>.<\/p>\n
    Przyk\u0142ad u\u017cycia<\/h2>\n
    Analizuj\u0105c przyk\u0142ad, zobaczymy jak dzia\u0142a mno\u017cenie macierzy:<\/p>\n
     =MACIERZ.ILOCZYN({1, 2; 3, 4}, {5; 6}) <\/code><\/pre>\n
    Macierz1 to macierz 2×2, a Macierz2 to macierz 2×1. Wynik mno\u017cenia to macierz 2×1, reprezentuj\u0105ca:<\/p>\n\n\n\n
    1*5 + 2*6<\/td>\n<\/tr>\n
    3*5 + 4*6<\/td>\n<\/tr>\n<\/table>\n
    Wynik oblicze\u0144 wynosi:<\/p>\n\n\n\n
    17<\/td>\n<\/tr>\n
    39<\/td>\n<\/tr>\n<\/table>\n
    Praktyczne zastosowania<\/h2>\n
    Zadanie 1: Obliczanie wynik\u00f3w eksperyment\u00f3w<\/h2>\n
    Za\u0142\u00f3\u017cmy, \u017ce prowadzimy serie eksperyment\u00f3w, aby zmierzy\u0107 wp\u0142yw dw\u00f3ch czynnik\u00f3w (A i B) na pewien system. Wyniki zapisane s\u0105 w macierzy 3×2, gdzie kolumny odpowiadaj\u0105 czynnikom, a wiersze poszczeg\u00f3lnym eksperymentom. U\u017cywaj\u0105c wektora wag dla czynnik\u00f3w [0.5, 2], obliczamy sumaryczny wp\u0142yw:<\/p>\n
     =MACIERZ.ILOCZYN({2, 3; 0, 7; 5, 1}, {0.5; 2}) <\/code><\/pre>\n
    Wynikowa macierz 3×1 przedstawia rezultaty dla ka\u017cdego eksperymentu:<\/p>\n\n\n\n\n
    2*0.5 + 3*2<\/td>\n<\/tr>\n
    0*0.5 + 7*2<\/td>\n<\/tr>\n
    5*0.5 + 1*2<\/td>\n<\/tr>\n<\/table>\n
    Co daje wyniki:<\/p>\n\n\n\n\n
    7.0<\/td>\n<\/tr>\n
    14.0<\/td>\n<\/tr>\n
    7.5<\/td>\n<\/tr>\n<\/table>\n
    Zadanie 2: Transformacja wsp\u00f3\u0142rz\u0119dnych<\/h2>\n
    Przyjmijmy, \u017ce chcemy obr\u00f3ci\u0107 zestaw punkt\u00f3w o 90 stopni wok\u00f3\u0142 pocz\u0105tku uk\u0142adu wsp\u00f3\u0142rz\u0119dnych. U\u017cywamy do tego macierzy rotacji R dla obrotu o 90 stopni w lewo, przedstawiona jako:<\/p>\n
     R = {0, -1; 1, 0} <\/code><\/pre>\n
    Reprezentuj\u0105c punkt [x, y] jako macierz 2×1 i stosuj\u0105c funkcj\u0119 mno\u017cenia macierzy, obliczamy nowe wsp\u00f3\u0142rz\u0119dne:<\/p>\n
     =MACIERZ.ILOCZYN({0, -1; 1, 0}, {x; y}) <\/code><\/pre>\n
    Wynikiem jest:<\/p>\n\n\n\n
    -y<\/td>\n<\/tr>\n
    x<\/td>\n<\/tr>\n<\/table>\n
    Co odpowiada obr\u00f3ceniu punktu o 90 stopni w lewo.<\/p>\n","protected":false},"excerpt":{"rendered":"
    Zwraca iloczyn macierzy dw\u00f3ch tablic<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[423],"tags":[],"class_list":["post-10534","post","type-post","status-publish","format-standard","hentry","category-matematyczne-i-trygonometryczne"],"acf":[],"yoast_head":"\nFunkcja MMULT (MACIERZ.ILOCZYN) w Excelu \u2013 jak dzia\u0142a, przyk\u0142ady u\u017cycia i sk\u0142adnia<\/title>\n<meta name=\"description\" content=\"Opis funkcji MMULT (MACIERZ.ILOCZYN) w Excelu. 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