{"id":757,"date":"2026-10-04T11:06:34","date_gmt":"2026-10-04T10:06:34","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/suma-wyznacznikow\/"},"modified":"2026-10-04T11:06:34","modified_gmt":"2026-10-04T10:06:34","slug":"suma-wyznacznikow","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/suma-wyznacznikow\/","title":{"rendered":"Kalkulator sumy wyznacznik\u00f3w"},"content":{"rendered":"<p><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\">Obliczanie sumy wyznacznik\u00f3w<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"757\" data-favorite-title=\"Suma wyznacznik\u00f3w\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/suma-wyznacznikow\/\" data-favorite-excerpt=\"Wprowad\u017a elementy dw\u00f3ch macierzy kwadratowych (2 \u00d7 2 lub 3 \u00d7 3), a ten kalkulator natychmiast obliczy oba wyznaczniki oraz ich sum\u0119 |A| + |B| wraz z obja\u015bnieniem wzor\u00f3w krok po kroku.\" 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\/suma-wyznacznikow\/\" method=\"POST\" class=\"calculator\" id=\"calculator-757\" data-post=\"757\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Rozmiar macierzy<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"rozmiar_macierzy\"><option value=\"2\">2 \u00d7 2<\/option><option value=\"3\">3 \u00d7 3<\/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-2\"><span class=\"font-semibold\">Macierz A<\/span><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">a\u2081\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-3\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2081\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-5\" hidden><label for=\"field-5\">a\u2081\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-5\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">a\u2082\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-6\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-7\"><label for=\"field-7\">a\u2082\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-7\" step=\"any\" value=\"5\"\/><\/div><div class=\"field\" id=\"input-8\" hidden><label for=\"field-8\">a\u2082\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-8\" step=\"any\" value=\"6\"\/><\/div><div class=\"field\" id=\"input-9\" hidden><label for=\"field-9\">a\u2083\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-9\" step=\"any\" value=\"7\"\/><\/div><div class=\"field\" id=\"input-10\" hidden><label for=\"field-10\">a\u2083\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-10\" step=\"any\" value=\"8\"\/><\/div><div class=\"field\" id=\"input-11\" hidden><label for=\"field-11\">a\u2083\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-11\" step=\"any\" value=\"9\"\/><\/div><div class=\"is-single field\" id=\"input-12\"><span class=\"font-semibold\">Macierz B<\/span><\/div><div class=\"field\" id=\"input-13\"><label for=\"field-13\">b\u2081\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-13\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-14\"><label for=\"field-14\">b\u2081\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-14\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-15\" hidden><label for=\"field-15\">b\u2081\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-15\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-16\"><label for=\"field-16\">b\u2082\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-16\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-17\"><label for=\"field-17\">b\u2082\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-17\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-18\" hidden><label for=\"field-18\">b\u2082\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-18\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-19\" hidden><label for=\"field-19\">b\u2083\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-19\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-20\" hidden><label for=\"field-20\">b\u2083\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-20\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-21\" hidden><label for=\"field-21\">b\u2083\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-21\" step=\"any\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"757\">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\" id=\"output-1\"><span class=\"field-title\"><span>|A|<\/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 class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>|B|<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/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 class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>|A| + |B|<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/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=\"757\" 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><br \/>\n<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\">Suma wyznacznik\u00f3w \u2013 przyk\u0142ad obliczenia<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"757\" data-favorite-title=\"Suma wyznacznik\u00f3w\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/suma-wyznacznikow\/\" data-favorite-excerpt=\"Wprowad\u017a elementy dw\u00f3ch macierzy kwadratowych (2 \u00d7 2 lub 3 \u00d7 3), a ten kalkulator natychmiast obliczy oba wyznaczniki oraz ich sum\u0119 |A| + |B| wraz z obja\u015bnieniem wzor\u00f3w krok po kroku.\" 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\/suma-wyznacznikow\/\" method=\"POST\" class=\"calculator\" id=\"calculator-757\" data-post=\"757\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Kwota pierwszej po\u017cyczki<\/label><div class=\"term\">L1<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"kwota_pierwszej_poyczki\" id=\"field-1\" aria-describedby=\"calc-unit-1 calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"100000\"\/><div class=\"unit\" id=\"calc-unit-1\">$<\/div><small id=\"calc-info-1\">Podaj kwot\u0119 pierwszej po\u017cyczki<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Oprocentowanie pierwszej po\u017cyczki<\/label><div class=\"term\">r1<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"oprocentowanie_pierwszej_poyczki\" id=\"field-2\" aria-describedby=\"calc-unit-2 calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" max=\"100\" value=\"4\"\/><div class=\"unit\" id=\"calc-unit-2\">%<\/div><small id=\"calc-info-2\">Podaj oprocentowanie pierwszej po\u017cyczki<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Kwota drugiej po\u017cyczki<\/label><div class=\"term\">L2<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"kwota_drugiej_poyczki\" id=\"field-3\" aria-describedby=\"calc-unit-3 calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"50000\"\/><div class=\"unit\" id=\"calc-unit-3\">$<\/div><small id=\"calc-info-3\">Podaj kwot\u0119 drugiej po\u017cyczki<\/small><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Oprocentowanie drugiej po\u017cyczki<\/label><div class=\"term\">r2<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"oprocentowanie_drugiej_poyczki\" id=\"field-4\" aria-describedby=\"calc-unit-4 calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" max=\"100\" value=\"6\"\/><div class=\"unit\" id=\"calc-unit-4\">%<\/div><small id=\"calc-info-4\">Podaj oprocentowanie drugiej po\u017cyczki<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"757\">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\" id=\"output-1\"><span class=\"field-title\"><span>\u0141\u0105czna stopa oprocentowania<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"%\"><\/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=\"757\" 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><\/p>\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 sum\u0119 wyznacznik\u00f3w<\/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\\[ |A| + |B| = \\det(A) + \\det(B) \\]\n\\[ \\det\\!\\begin{bmatrix} a &amp; b \\\\ c &amp; d \\end{bmatrix} = ad - bc \\]\n\\[ \\det\\!\\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{bmatrix} = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(A\\) i \\(B\\) \u2014 macierze kwadratowe tego samego rozmiaru (2 \u00d7 2 lub 3 \u00d7 3)<\/li>\n<li>\\(\\det(A)\\), zapisywany \\(|A|\\) \u2014 wyznacznik macierzy \\(A\\)<\/li>\n<li>\\(\\det(B)\\), zapisywany \\(|B|\\) \u2014 wyznacznik macierzy \\(B\\)<\/li>\n<\/ul>\n<p>Pierwszy wz\u00f3r opisuje sam\u0105 operacj\u0119: ka\u017cdy wyznacznik jest obliczany niezale\u017cnie, a dwa wyniki skalarne s\u0105 nast\u0119pnie do siebie dodawane. Drugi i trzeci wz\u00f3r pokazuj\u0105, jak oblicza si\u0119 pojedynczy wyznacznik dla macierzy 2 \u00d7 2 i 3 \u00d7 3 (rozwini\u0119cie wzgl\u0119dem pierwszego wiersza metod\u0105 dope\u0142nie\u0144 algebraicznych).<\/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>Suma wyznacznik\u00f3w \u2013 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> dwie macierze 3 \u00d7 3 z domy\u015blnymi warto\u015bciami kalkulatora:<\/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} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix} \\]<\/p>\n<p>\\[ \\det(A) = 1(5 \\cdot 9 - 6 \\cdot 8) - 2(4 \\cdot 9 - 6 \\cdot 7) + 3(4 \\cdot 8 - 5 \\cdot 7) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0 \\]<\/p>\n<p>\\[ \\det(B) = 1(1 \\cdot 1 - 0 \\cdot 0) - 0(0 \\cdot 1 - 0 \\cdot 0) + 0(0 \\cdot 0 - 1 \\cdot 0) = 1 \\]<\/p>\n<p>\\[ |A| + |B| = 0 + 1 = 1 \\]<\/p>\n<p><strong>Wynik:<\/strong> \\(\\det(A) = 0\\), \\(\\det(B) = 1\\), a suma wyznacznik\u00f3w wynosi \\(1\\). Zwr\u00f3\u0107 uwag\u0119, \u017ce macierz \\(A\\) jest osobliwa (jej wiersze s\u0105 liniowo zale\u017cne), dlatego jej wyznacznik jest r\u00f3wny zeru.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wybierz <strong>rozmiar macierzy<\/strong> \u2014 2 \u00d7 2 albo 3 \u00d7 3 \u2014 z listy rozwijanej u g\u00f3ry. Pola dla macierzy 3 \u00d7 3 pojawi\u0105 si\u0119 lub znikn\u0105 automatycznie.<\/li>\n<li>Wpisz elementy <strong>macierzy A<\/strong> w polach oznaczonych a\u2081\u2081 do a\u2083\u2083 (wiersz po wierszu).<\/li>\n<li>Wpisz elementy <strong>macierzy B<\/strong> w polach oznaczonych b\u2081\u2081 do b\u2083\u2083.<\/li>\n<li>Wyniki aktualizuj\u0105 si\u0119 natychmiast: <strong>|A|<\/strong>, <strong>|B|<\/strong> oraz suma <strong>|A| + |B|<\/strong>.<\/li>\n<\/ol>\n<h2>Czym jest suma wyznacznik\u00f3w?<\/h2>\n<p>Wyznacznik to pojedyncza liczba obliczana na podstawie macierzy kwadratowej. Koduje ona kluczowe w\u0142asno\u015bci przekszta\u0142cenia liniowego reprezentowanego przez macierz: czy przekszta\u0142cenie jest odwracalne, jak skaluje pola lub obj\u0119to\u015bci oraz jak zmienia orientacj\u0119 przestrzeni. Wyznacznik r\u00f3wny zeru oznacza, \u017ce macierz jest osobliwa \u2014 \u201esp\u0142aszcza\u201d przestrze\u0144 i nie ma macierzy odwrotnej.<\/p>\n<p>Suma wyznacznik\u00f3w to po prostu dodawanie skalarne \\(\\det(A) + \\det(B)\\). Ka\u017cdy wyznacznik jest wyliczany osobno, a \u0142\u0105czone s\u0105 tylko dwie otrzymane liczby. Jest to typowe \u0107wiczenie na kursach algebry liniowej, pojawia si\u0119 te\u017c w dowodach i wzorach, gdzie trzeba po\u0142\u0105czy\u0107 wyznaczniki kilku macierzy \u2014 na przyk\u0142ad przy badaniu zale\u017cno\u015bci liniowej, wyliczaniu wyra\u017ce\u0144 z macierzy blokowych czy \u0142\u0105czeniu wielomian\u00f3w charakterystycznych.<\/p>\n<p>Kluczowa uwaga: <strong>suma wyznacznik\u00f3w zasadniczo nie jest r\u00f3wna wyznacznikowi sumy<\/strong>. W zapisie symbolicznym \\(\\det(A + B) \\neq \\det(A) + \\det(B)\\) w prawie wszystkich przypadkach. Wyznacznik jest funkcj\u0105 nieliniow\u0105 i multiplikatywn\u0105 \u2014 spe\u0142nia \\(\\det(AB) = \\det(A)\\,\\det(B)\\), co mo\u017cesz sprawdzi\u0107 naszym kalkulatorem <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/mnozenie-wyznacznikow\/\">Mno\u017cenie wyznacznik\u00f3w<\/a> \u2014 ale nie rozdziela si\u0119 na dodawanie macierzy. Je\u015bli potrzebujesz wyznacznika macierzy \\(A + B\\), najpierw dodaj macierze element po elemencie kalkulatorem <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/dodawanie-macierzy\/\">Dodawanie macierzy<\/a>, a nast\u0119pnie oblicz wyznacznik wyniku.<\/p>\n<p>Interpretacja wyniku jest prosta: <strong>|A|<\/strong> i <strong>|B|<\/strong> to dwa pojedyncze wyznaczniki, a <strong>|A| + |B|<\/strong> to ich zwyk\u0142a suma arytmetyczna, kt\u00f3ra mo\u017ce by\u0107 dodatnia, ujemna lub zerowa. Suma r\u00f3wna zero nie oznacza, \u017ce kt\u00f3ra\u015b macierz jest osobliwa \u2014 jeden wyznacznik mo\u017ce po prostu zniwelowa\u0107 drugi.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Wy\u0142\u0105cznie macierze kwadratowe tego samego rozmiaru.<\/strong> Wyznaczniki s\u0105 zdefiniowane tylko dla macierzy kwadratowych, a to narz\u0119dzie przyjmuje dane 2 \u00d7 2 i 3 \u00d7 3. Macierze nie musz\u0105 by\u0107 w \u017caden inny spos\u00f3b ze sob\u0105 powi\u0105zane.<\/li>\n<li><strong>Znaki maj\u0105 znaczenie.<\/strong> Wyznacznik mo\u017ce by\u0107 ujemny. Zamiana dw\u00f3ch wierszy lub dw\u00f3ch kolumn zmienia znak na przeciwny, dlatego sprawd\u017a dok\u0142adnie kolejno\u015b\u0107 wpisywanych warto\u015bci (najpierw wiersz, potem kolumna: a\u2082\u2081 to wiersz 2, kolumna 1).<\/li>\n<li><strong>Zaokr\u0105glanie.<\/strong> Kalkulator zaokr\u0105gla wy\u015bwietlane wyniki, wi\u0119c bardzo du\u017ce warto\u015bci lub liczby z wieloma miejscami po przecinku mog\u0105 straci\u0107 nieco precyzji w pokazanym wyniku.<\/li>\n<li><strong>Macierze osobliwe.<\/strong> Je\u015bli wyznacznik oka\u017ce si\u0119 r\u00f3wny 0, macierz nie ma odwrotno\u015bci, a jej wiersze (i kolumny) s\u0105 liniowo zale\u017cne. Aby bli\u017cej przyjrze\u0107 si\u0119 takiej macierzy, wypr\u00f3buj kalkulator <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/rzad-macierzy-3x3\/\">Rz\u0105d macierzy 3\u00d73<\/a> lub kalkulator <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/macierz-odwrotna-3x3\/\">Macierz odwrotna 3\u00d73<\/a>.<\/li>\n<li><strong>Potrzebujesz odejmowania?<\/strong> Aby uzyska\u0107 r\u00f3\u017cnic\u0119 \\(\\det(A) - \\det(B)\\), mo\u017cesz wpisa\u0107 \\(-B\\) jako drug\u0105 macierz, poniewa\u017c \\(\\det(-B) = -\\det(B)\\) dla wymiaru 3 \u00d7 3 (a og\u00f3lnie \\(\\det(-B) = (-1)^n\\det(B)\\) dla macierzy \\(n \\times n\\)). Odejmowanie macierzy element po elemencie obs\u0142uguje nasz kalkulator <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/odejmowanie-macierzy\/\">Odejmowanie dw\u00f3ch macierzy<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Wprowad\u017a elementy dw\u00f3ch macierzy kwadratowych (2 \u00d7 2 lub 3 \u00d7 3), a ten kalkulator natychmiast obliczy oba wyznaczniki oraz ich sum\u0119 |A| + |B| wraz z obja\u015bnieniem wzor\u00f3w krok po kroku.<\/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-757","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\/757","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=757"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/757\/revisions"}],"predecessor-version":[{"id":2129,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/757\/revisions\/2129"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=757"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=757"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=757"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}