{"id":854,"date":"2026-10-04T17:33:47","date_gmt":"2026-10-04T16:33:47","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/mnozenie-macierzy\/"},"modified":"2026-10-04T17:33:47","modified_gmt":"2026-10-04T16:33:47","slug":"mnozenie-macierzy","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/mnozenie-macierzy\/","title":{"rendered":"Kalkulator mno\u017cenia macierzy"},"content":{"rendered":"<style id=\"wpcoderpro-inline-815\">.calculator input[readonly] {\r\n\toutline-color: oklch(70.4% 0.191 22.216);\r\n}<\/style><section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Rozwi\u0105zywanie mno\u017cenia macierzy<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"854\" data-favorite-title=\"Mno\u017cenie macierzy\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/mnozenie-macierzy\/\" data-favorite-excerpt=\"Pomn\u00f3\u017c natychmiast dwie macierze o zgodnych rozmiarach (do 3 \u00d7 3). Wybierz wymiary macierzy A i B, wprowad\u017a ich elementy i otrzymaj pe\u0142n\u0105 macierz iloczynu A \u00d7 B.\" aria-label=\"Dodaj do ulubionych\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/pl\/matematyka\/mnozenie-macierzy\/\" method=\"POST\" class=\"calculator\" id=\"calculator-854\" data-post=\"854\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Rozmiar macierzy A<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"rozmiar_macierzy_a\"><option value=\"3x3\">3 \u00d7 3<\/option><option value=\"3x2\">3 \u00d7 2<\/option><option value=\"2x3\">2 \u00d7 3<\/option><option value=\"3x1\">3 \u00d7 1<\/option><option value=\"1x3\">1 \u00d7 3<\/option><option value=\"2x2\">2 \u00d7 2<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field is-single\" id=\"input-2\"><label for=\"field-2\">Rozmiar macierzy B<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"rozmiar_macierzy_b\"><option value=\"3x3\">3 \u00d7 3<\/option><option value=\"3x2\">3 \u00d7 2<\/option><option value=\"2x3\">2 \u00d7 3<\/option><option value=\"3x1\">3 \u00d7 1<\/option><option value=\"1x3\">1 \u00d7 3<\/option><option value=\"2x2\">2 \u00d7 2<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"is-single field\" id=\"input-3\"><span class=\"font-semibold\">Macierz A<\/span><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">A11<\/label><input type=\"number\" name=\"a11\" id=\"field-4\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">A12<\/label><input type=\"number\" name=\"a12\" id=\"field-5\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">A13<\/label><input type=\"number\" name=\"a13\" id=\"field-6\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-7\"><label for=\"field-7\">A21<\/label><input type=\"number\" name=\"a21\" id=\"field-7\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-8\"><label for=\"field-8\">A22<\/label><input type=\"number\" name=\"a22\" id=\"field-8\" step=\"any\" value=\"5\"\/><\/div><div class=\"field\" id=\"input-9\"><label for=\"field-9\">A23<\/label><input type=\"number\" name=\"a23\" id=\"field-9\" step=\"any\" value=\"6\"\/><\/div><div class=\"field\" id=\"input-10\"><label for=\"field-10\">A31<\/label><input type=\"number\" name=\"a31\" id=\"field-10\" step=\"any\" value=\"7\"\/><\/div><div class=\"field\" id=\"input-11\"><label for=\"field-11\">A32<\/label><input type=\"number\" name=\"a32\" id=\"field-11\" step=\"any\" value=\"8\"\/><\/div><div class=\"field\" id=\"input-12\"><label for=\"field-12\">A33<\/label><input type=\"number\" name=\"a33\" id=\"field-12\" step=\"any\" value=\"9\"\/><\/div><div class=\"is-single field\" id=\"input-13\"><span class=\"font-semibold\">Macierz B<\/span><\/div><div class=\"field\" id=\"input-14\"><label for=\"field-14\">B11<\/label><input type=\"number\" name=\"b11\" id=\"field-14\" step=\"any\" value=\"9\"\/><\/div><div class=\"field\" id=\"input-15\"><label for=\"field-15\">B12<\/label><input type=\"number\" name=\"b12\" id=\"field-15\" step=\"any\" value=\"8\"\/><\/div><div class=\"field\" id=\"input-16\"><label for=\"field-16\">B13<\/label><input type=\"number\" name=\"b13\" id=\"field-16\" step=\"any\" value=\"7\"\/><\/div><div class=\"field\" id=\"input-17\"><label for=\"field-17\">B21<\/label><input type=\"number\" name=\"b21\" id=\"field-17\" step=\"any\" value=\"6\"\/><\/div><div class=\"field\" id=\"input-18\"><label for=\"field-18\">B22<\/label><input type=\"number\" name=\"b22\" id=\"field-18\" step=\"any\" value=\"5\"\/><\/div><div class=\"field\" id=\"input-19\"><label for=\"field-19\">B23<\/label><input type=\"number\" name=\"b23\" id=\"field-19\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-20\"><label for=\"field-20\">B31<\/label><input type=\"number\" name=\"b31\" id=\"field-20\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-21\"><label for=\"field-21\">B32<\/label><input type=\"number\" name=\"b32\" id=\"field-21\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-22\"><label for=\"field-22\">B33<\/label><input type=\"number\" name=\"b33\" id=\"field-22\" step=\"any\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"854\">Oblicz<\/button><button type=\"reset\">Wyczy\u015b\u0107<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Obliczaj automatycznie<\/label><small>Po w\u0142\u0105czeniu wynik aktualizuje si\u0119 automatycznie po zmianie dowolnej warto\u015bci.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Twoje wyniki<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Wynik mno\u017cenia macierzy (A \u00d7 B)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Kopiuj link do tego obliczenia<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Zostaw komentarz<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"854\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Jak oceniasz ten wynik?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Lubi\u0119 to <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Pomocne <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Niejasne <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Rozczarowanie <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Niedok\u0142adne <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Wz\u00f3r na mno\u017cenie macierzy<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Wz\u00f3r<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ C_{ij} = \\sum_{k=1}^{n} A_{ik} B_{kj} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(C_{ij}\\) \u2014 element w wierszu \\(i\\) i kolumnie \\(j\\) macierzy wynikowej<\/li>\n<li>\\(A_{ik}\\) \u2014 element w wierszu \\(i\\) i kolumnie \\(k\\) macierzy A<\/li>\n<li>\\(B_{kj}\\) \u2014 element w wierszu \\(k\\) i kolumnie \\(j\\) macierzy B<\/li>\n<li>\\(n\\) \u2014 wsp\u00f3lny wymiar: liczba kolumn macierzy A, kt\u00f3ra musi by\u0107 r\u00f3wna liczbie wierszy macierzy B<\/li>\n<\/ul>\n<p>Ka\u017cdy element iloczynu to suma iloczyn\u00f3w odpowiednich wpis\u00f3w z wiersza macierzy A i kolumny macierzy B \u2014 klasyczna regu\u0142a \u201ewiersz razy kolumna\u201d.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Mno\u017cenie macierzy \u2013 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> pomn\u00f3\u017c dwie macierze 3 \u00d7 3 (s\u0105 to domy\u015blne warto\u015bci kalkulatora, wi\u0119c mo\u017cesz odtworzy\u0107 powy\u017cszy wynik).<\/p>\n<p>\\[ A = \\begin{bmatrix} 1 &amp; 2 &amp; 3 \\\\ 4 &amp; 5 &amp; 6 \\\\ 7 &amp; 8 &amp; 9 \\end{bmatrix}, \\quad B = \\begin{bmatrix} 9 &amp; 8 &amp; 7 \\\\ 6 &amp; 5 &amp; 4 \\\\ 3 &amp; 2 &amp; 1 \\end{bmatrix} \\]<\/p>\n<p>We\u017a wiersz 1 macierzy A i kolumn\u0119 1 macierzy B: \\( 1 \\cdot 9 + 2 \\cdot 6 + 3 \\cdot 3 = 30 \\). Powtarzaj\u0105c to dla ka\u017cdej pary wiersz\u2013kolumna, otrzymujemy:<\/p>\n<p>\\[ C = A \\times B = \\begin{bmatrix} 30 &amp; 24 &amp; 18 \\\\ 84 &amp; 69 &amp; 54 \\\\ 138 &amp; 114 &amp; 90 \\end{bmatrix} \\]<\/p>\n<p><strong>Wynik:<\/strong> macierz 3 \u00d7 3, kt\u00f3rej pierwszy element to 30, drugi 24 i tak dalej.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Ustaw <strong>rozmiar macierzy A<\/strong> (np. 3 \u00d7 3) i <strong>rozmiar macierzy B<\/strong> za pomoc\u0105 list rozwijanych. Liczba kolumn macierzy A musi by\u0107 r\u00f3wna liczbie wierszy macierzy B \u2014 w przeciwnym razie kalkulator zg\u0142osi niezgodno\u015b\u0107 rozmiar\u00f3w.<\/li>\n<li>Wprowad\u017a elementy <strong>macierzy A<\/strong> w pola A11, A12, \u2026 Pola poza wybranym rozmiarem staj\u0105 si\u0119 tylko do odczytu i s\u0105 ignorowane.<\/li>\n<li>Wprowad\u017a elementy <strong>macierzy B<\/strong> w pola B11, B12, \u2026<\/li>\n<li>Odczytaj macierz iloczynu <strong>A \u00d7 B<\/strong> z tabeli wynik\u00f3w. Jej wymiary to (wiersze A) \u00d7 (kolumny B).<\/li>\n<\/ol>\n<h2>Czym jest mno\u017cenie macierzy?<\/h2>\n<p>Mno\u017cenie macierzy to operacja dwuargumentowa, kt\u00f3ra \u0142\u0105czy dwie macierze w now\u0105 macierz. W przeciwie\u0144stwie do zwyk\u0142ego mno\u017cenia liczb <em>nie<\/em> odbywa si\u0119 ono element po elemencie: ka\u017cdy wpis wyniku jest iloczynem skalarnym wiersza pierwszej macierzy i kolumny drugiej. Macierz A o wymiarach \\(m \\times n\\) mo\u017cna pomno\u017cy\u0107 przez macierz B o wymiarach \\(n \\times p\\), a wynik ma rozmiar \\(m \\times p\\).<\/p>\n<p>Kolejno\u015b\u0107 mno\u017cenia ma znaczenie. Og\u00f3lnie \\(AB \\neq BA\\), a cz\u0119sto tylko jeden z dw\u00f3ch iloczyn\u00f3w jest w og\u00f3le okre\u015blony. Kiedy obliczasz \\(A \\times B\\), mno\u017cysz <em>wiersze A przez kolumny B<\/em> \u2014 zamiana macierzy daje inn\u0105 (lub niemo\u017cliw\u0105) operacj\u0119.<\/p>\n<p>Iloczyny macierzy pojawiaj\u0105 si\u0119 wsz\u0119dzie w matematyce stosowanej: przy przekszta\u0142caniu punkt\u00f3w i obiekt\u00f3w w grafice komputerowej, \u0142\u0105czeniu przekszta\u0142ce\u0144 w robotyce, rozwi\u0105zywaniu uk\u0142ad\u00f3w r\u00f3wna\u0144 liniowych oraz w warstwach sieci neuronowych w uczeniu maszynowym. Pomno\u017cenie macierzy przez sam\u0105 siebie daje macierz do kwadratu, kt\u00f3r\u0105 mo\u017cesz obliczy\u0107 za pomoc\u0105 <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/kwadrat-macierzy\/\">kalkulatora kwadratu macierzy<\/a>.<\/p>\n<h2>Interpretacja wyniku<\/h2>\n<p>Tabela wynik\u00f3w pokazuje macierz iloczynu \\(C\\) o \\(m\\) wierszach i \\(p\\) kolumnach. Ka\u017cdy element \\(C_{ij}\\) powsta\u0142 przez pomno\u017cenie \\(i\\)-tego wiersza macierzy A element po elemencie przez \\(j\\)-t\u0105 kolumn\u0119 macierzy B i zsumowanie wynik\u00f3w. Je\u015bli kt\u00f3rykolwiek wiersz A zostanie pomno\u017cony w z\u0142ej kolejno\u015bci, element b\u0119dzie b\u0142\u0119dny \u2014 i w\u0142a\u015bnie przed takimi pomy\u0142kami chroni ten kalkulator.<\/p>\n<p>Je\u015bli potrzebujesz po\u0142\u0105czy\u0107 macierze wpis po wpisie, u\u017cyj zamiast tego kalkulator\u00f3w <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/dodawanie-macierzy\/\">dodawania macierzy<\/a> lub <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/odejmowanie-macierzy\/\">odejmowania macierzy<\/a> \u2014 te operacje dzia\u0142aj\u0105 element po elemencie i wymagaj\u0105 identycznych wymiar\u00f3w.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Ograniczenia rozmiaru:<\/strong> ten kalkulator obs\u0142uguje macierze do 3 \u00d7 3. Wi\u0119ksze iloczyny oblicza si\u0119 tak samo, ale trzeba to zrobi\u0107 w programie lub r\u0119cznie.<\/li>\n<li><strong>Regu\u0142a zgodno\u015bci:<\/strong> mno\u017cenie jest okre\u015blone tylko wtedy, gdy liczba kolumn A = liczbie wierszy B. Macierz 2 \u00d7 3 razy macierz 3 \u00d7 2 dzia\u0142a (wynik 2 \u00d7 2); macierz 3 \u00d7 2 razy macierz 2 \u00d7 3 r\u00f3wnie\u017c dzia\u0142a (wynik 3 \u00d7 3) \u2014 ale daj\u0105 r\u00f3\u017cne odpowiedzi.<\/li>\n<li><strong>Nieprzemienno\u015b\u0107:<\/strong> poniewa\u017c og\u00f3lnie \\(AB \\neq BA\\), zawsze sprawdzaj, kt\u00f3ra macierz wyst\u0119puje pierwsza w Twoim zadaniu.<\/li>\n<li><strong>Skr\u00f3t z wyznacznikiem:<\/strong> dla macierzy kwadratowych \\(\\det(AB) = \\det(A) \\cdot \\det(B)\\). Mo\u017cesz to sprawdzi\u0107 za pomoc\u0105 kalkulatora <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/mnozenie-wyznacznikow\/\">mno\u017cenia wyznacznik\u00f3w<\/a>.<\/li>\n<li><strong>Zastosowania:<\/strong> je\u015bli iloczyn ma wej\u015b\u0107 do dalszych oblicze\u0144 algebry liniowej \u2014 na przyk\u0142ad znajdowania macierzy odwrotnej \u2014 <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/macierz-odwrotna-3x3\/\">kalkulator macierzy odwrotnej 3\u00d73<\/a> obejmuje nast\u0119pny krok.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Pomn\u00f3\u017c natychmiast dwie macierze o zgodnych rozmiarach (do 3 \u00d7 3). Wybierz wymiary macierzy A i B, wprowad\u017a ich elementy i otrzymaj pe\u0142n\u0105 macierz iloczynu A \u00d7 B.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[6],"tags":[51],"class_list":["post-854","post","type-post","status-publish","format-standard","hentry","category-matematyka","tag-macierze"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/854","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/comments?post=854"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/854\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=854"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=854"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=854"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}