{"id":4183,"date":"2026-11-02T10:20:00","date_gmt":"2026-11-02T10:20:00","guid":{"rendered":"https:\/\/wpcalc.com\/tr\/matematik\/matris-determinant-hesaplama\/"},"modified":"2026-11-02T10:20:00","modified_gmt":"2026-11-02T10:20:00","slug":"matris-determinant-hesaplama","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/tr\/matematik\/matris-determinant-hesaplama\/","title":{"rendered":"Matris Determinant Hesaplama"},"content":{"rendered":"<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\">Kare bir matrisin determinant\u0131n\u0131 hesaplay\u0131n<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"4183\" data-favorite-title=\"Determinant\" data-favorite-url=\"https:\/\/wpcalc.com\/tr\/matematik\/matris-determinant-hesaplama\/\" data-favorite-excerpt=\"2\u00d72&#039;den 6\u00d76&#039;ya kadar herhangi bir kare matrisin determinant\u0131n\u0131 tam olarak hesaplay\u0131n. De\u011ferleri tam say\u0131, ondal\u0131kl\u0131 say\u0131 veya kesir olarak girin; kesin sonu\u00e7, ondal\u0131k de\u011fer, tekil (sing\u00fcler) olup olmad\u0131\u011f\u0131 denetimi ve ad\u0131m ad\u0131m \u00e7\u00f6z\u00fcm\u00fc an\u0131nda al\u0131n.\" aria-label=\"Favorilere ekle\" 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\/tr\/matematik\/matris-determinant-hesaplama\/\" method=\"POST\" class=\"calculator\" id=\"calculator-4183\" data-post=\"4183\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Giri\u015f alanlar\u0131<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Matris boyutu<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"matris_boyutu\"><option value=\"2\">2 \u00d7 2<\/option><option value=\"3\" selected>3 \u00d7 3<\/option><option value=\"4\">4 \u00d7 4<\/option><option value=\"5\">5 \u00d7 5<\/option><option value=\"6\">6 \u00d7 6<\/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\" id=\"input-2\"><label for=\"field-2\">Izgaray\u0131 doldurma y\u00f6ntemi<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"izgaray_doldurma_yntemi\"><option value=\"custom\" selected>A\u015fa\u011f\u0131ya girilen de\u011ferleri kullan<\/option><option value=\"random\">Rastgele tam say\u0131 matrisi<\/option><option value=\"identity\">Birim matris<\/option><option value=\"zeros\">Temizle (tamam\u0131 s\u0131f\u0131r)<\/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\">Sat\u0131r 1<\/span><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2081\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-4\" aria-describedby=\"calc-info-4\" placeholder=\"0\" step=\"any\" value=\"2\"\/><small id=\"calc-info-4\">Tam say\u0131lar, ondal\u0131kl\u0131 say\u0131lar (\u00f6rn. 1,5) veya kesirler (\u00f6rn. 3\/4). Bo\u015f h\u00fccreler 0 kabul edilir.<\/small><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">a\u2081\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-5\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">a\u2081\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-6\" placeholder=\"0\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-7\" hidden><label for=\"field-7\">a\u2081\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-7\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-8\" hidden><label for=\"field-8\">a\u2081\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-8\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-9\" hidden><label for=\"field-9\">a\u2081\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-9\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-10\"><span class=\"font-semibold\">Sat\u0131r 2<\/span><\/div><div class=\"field\" id=\"input-11\"><label for=\"field-11\">a\u2082\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-11\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-12\"><label for=\"field-12\">a\u2082\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-12\" placeholder=\"0\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-13\"><label for=\"field-13\">a\u2082\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-13\" placeholder=\"0\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-14\" hidden><label for=\"field-14\">a\u2082\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-14\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-15\" hidden><label for=\"field-15\">a\u2082\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-15\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-16\" hidden><label for=\"field-16\">a\u2082\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-16\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-17\"><span class=\"font-semibold\">Sat\u0131r 3<\/span><\/div><div class=\"field\" id=\"input-18\"><label for=\"field-18\">a\u2083\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-18\" placeholder=\"0\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-19\"><label for=\"field-19\">a\u2083\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-19\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-20\"><label for=\"field-20\">a\u2083\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-20\" placeholder=\"0\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-21\" hidden><label for=\"field-21\">a\u2083\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-21\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-22\" hidden><label for=\"field-22\">a\u2083\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-22\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-23\" hidden><label for=\"field-23\">a\u2083\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-23\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-24\" hidden><span class=\"font-semibold\">Sat\u0131r 4<\/span><\/div><div class=\"field\" id=\"input-25\" hidden><label for=\"field-25\">a\u2084\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-25\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-26\" hidden><label for=\"field-26\">a\u2084\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-26\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-27\" hidden><label for=\"field-27\">a\u2084\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-27\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-28\" hidden><label for=\"field-28\">a\u2084\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-28\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-29\" hidden><label for=\"field-29\">a\u2084\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-29\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-30\" hidden><label for=\"field-30\">a\u2084\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-30\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-31\" hidden><span class=\"font-semibold\">Sat\u0131r 5<\/span><\/div><div class=\"field\" id=\"input-32\" hidden><label for=\"field-32\">a\u2085\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-32\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-33\" hidden><label for=\"field-33\">a\u2085\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-33\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-34\" hidden><label for=\"field-34\">a\u2085\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-34\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-35\" hidden><label for=\"field-35\">a\u2085\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-35\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-36\" hidden><label for=\"field-36\">a\u2085\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-36\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-37\" hidden><label for=\"field-37\">a\u2085\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-37\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-38\" hidden><span class=\"font-semibold\">Sat\u0131r 6<\/span><\/div><div class=\"field\" id=\"input-39\" hidden><label for=\"field-39\">a\u2086\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-39\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-40\" hidden><label for=\"field-40\">a\u2086\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-40\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-41\" hidden><label for=\"field-41\">a\u2086\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-41\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-42\" hidden><label for=\"field-42\">a\u2086\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-42\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-43\" hidden><label for=\"field-43\">a\u2086\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-43\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-44\" hidden><label for=\"field-44\">a\u2086\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-44\" placeholder=\"0\" step=\"any\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Yeniden hesapla\" id=\"calculate-button\" data-post=\"4183\">Hesapla<\/button><button type=\"reset\">S\u0131f\u0131rla<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Otomatik hesapla<\/label><small>Etkinle\u015ftirilirse, herhangi bir de\u011feri de\u011fi\u015ftirdi\u011finizde sonu\u00e7 otomatik olarak g\u00fcncellenir.<\/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> Determinant sonu\u00e7lar\u0131<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Determinant (kesin de\u011fer)<\/span><span class=\"result-term\">det A<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Determinant (ondal\u0131k)<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>Matris tekil mi?<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-4\"><span class=\"field-title\"><span>\u00c7\u00f6z\u00fcm ad\u0131mlar\u0131<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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\">Bu hesaplaman\u0131n ba\u011flant\u0131s\u0131n\u0131 kopyala<\/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>Yorum b\u0131rak<\/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=\"4183\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Bu sonu\u00e7 hakk\u0131nda ne d\u00fc\u015f\u00fcn\u00fcyorsunuz?<\/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\">Be\u011fendim <\/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\">Faydal\u0131 <\/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\">Kafam kar\u0131\u015ft\u0131 <\/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\">Hayal k\u0131r\u0131kl\u0131\u011f\u0131 <\/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\">Hatal\u0131 <\/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>Determinant Form\u00fcl\u00fc<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Form\u00fcl<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[\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\\det A = \\sum_{j=1}^{n} (-1)^{1+j}\\, a_{1j}\\, M_{1j}\n\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Burada:<\/strong><\/p>\n<ul>\n<li>\\(a_{ij}\\) \u2014 kare matris \\(A\\)'n\u0131n \\(i\\). sat\u0131r\u0131ndaki, \\(j\\). s\u00fctunundaki eleman<\/li>\n<li>\\(M_{1j}\\) \u2014 <em>min\u00f6r<\/em>: 1. sat\u0131r ile \\(j\\). s\u00fctun \u00e7\u0131kar\u0131ld\u0131ktan sonra kalan alt matrisin determinant\u0131<\/li>\n<li>\\((-1)^{1+j}M_{1j}\\) \u2014 \\(a_{1j}\\) eleman\u0131n\u0131n <em>kofakt\u00f6r\u00fc<\/em><\/li>\n<\/ul>\n<p>\u0130lk sat\u0131r 2\u00d72 matrisler i\u00e7in k\u00f6\u015fegen kural\u0131, ikincisi 3\u00d73 matrisler i\u00e7in ilk sat\u0131r boyunca kofakt\u00f6r (Laplace) a\u00e7\u0131l\u0131m\u0131, son sat\u0131r ise herhangi bir \\(n \\times n\\) matris i\u00e7in ge\u00e7erli genel a\u00e7\u0131l\u0131md\u0131r.<\/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>Determinant \u2013 Hesaplama \u00d6rne\u011fi<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>\u00d6rnek:<\/strong> a\u015fa\u011f\u0131daki 3\u00d73 matrisin determinant\u0131n\u0131 bulal\u0131m<\/p>\n<p>\\[ A = \\begin{bmatrix} 2 &amp; 1 &amp; 3 \\\\ 1 &amp; 0 &amp; 2 \\\\ 4 &amp; 1 &amp; 3 \\end{bmatrix} \\]<\/p>\n<p>\\[ M_{11} = 0 \\cdot 3 - 2 \\cdot 1 = -2, \\quad M_{12} = 1 \\cdot 3 - 2 \\cdot 4 = -5, \\quad M_{13} = 1 \\cdot 1 - 0 \\cdot 4 = 1 \\]<\/p>\n<p>\\[ \\det A = 2(-2) - 1(-5) + 3(1) = -4 + 5 + 3 \\]<\/p>\n<p><strong>Sonu\u00e7:<\/strong> \\(\\det A = 4\\). Determinant s\u0131f\u0131rdan farkl\u0131 oldu\u011fu i\u00e7in matris tersinirdir. Arac\u0131n varsay\u0131lan de\u011ferleriyle hesapla d\u00fc\u011fmesine basarak bu sonucu kendiniz de do\u011frulayabilirsiniz.<\/p>\n<p>\n\r\n<\/section>\n<h2>Determinant Hesaplama Arac\u0131 Nas\u0131l Kullan\u0131l\u0131r?<\/h2>\n<ol>\n<li><strong>Matris boyutu<\/strong> a\u00e7\u0131l\u0131r men\u00fcs\u00fcn\u00fc a\u00e7\u0131n ve 2 \u00d7 2, 3 \u00d7 3, 4 \u00d7 4, 5 \u00d7 5 veya 6 \u00d7 6 se\u00e7ene\u011fini belirleyin. Yaln\u0131zca se\u00e7ilen boyuta ait alanlar kullan\u0131l\u0131r.<\/li>\n<li>Matrisi nas\u0131l girece\u011finizi <strong>Izgaraay\u0131 doldur<\/strong> a\u00e7\u0131l\u0131r men\u00fcs\u00fcnden se\u00e7in: <em>A\u015fa\u011f\u0131da girilen de\u011ferleri kullan<\/em>, <em>Rastgele tam say\u0131l\u0131 matris<\/em> (her ge\u00e7i\u015fte yeni bir matris \u00fcretilir), <em>Birim matris<\/em> veya <em>Temizle (t\u00fcm\u00fc s\u0131f\u0131r)<\/em>.<\/li>\n<li>Elemanlar\u0131n\u0131z\u0131 a<sub>11<\/sub>, a<sub>12<\/sub>, \u2026 alanlar\u0131na yaz\u0131n. Tam say\u0131lar, 1,5 gibi ondal\u0131kl\u0131 say\u0131lar ve 3\/4 gibi kesirler kabul edilir. Bo\u015f h\u00fccreler 0 kabul edilir.<\/li>\n<li>Sonu\u00e7lar\u0131 okuyun: <strong>Determinant (kesin)<\/strong> cevab\u0131 tam kesir veya tam say\u0131 olarak verir, <strong>Determinant (ondal\u0131k)<\/strong> ondal\u0131k g\u00f6sterimini verir, <strong>Matris tekil mi?<\/strong> matrisin tersinir olup olmad\u0131\u011f\u0131n\u0131 s\u00f6yler ve <strong>\u00c7\u00f6z\u00fcm ad\u0131mlar\u0131<\/strong> t\u00fcm i\u015flemin detaylar\u0131n\u0131 g\u00f6sterir.<\/li>\n<\/ol>\n<h2>Bir Matrisin Determinant\u0131 Nedir?<\/h2>\n<p>Determinant, bir kare matrisin \u00f6nemli bir geometrik ve cebirsel \u00f6zelli\u011fini tek bir say\u0131da \u00f6zetler. Geometrik olarak, bir \\(n \\times n\\) matrisin determinant\u0131n\u0131n mutlak de\u011feri, matrisin temsil etti\u011fi do\u011frusal d\u00f6n\u00fc\u015f\u00fcm\u00fcn hacim \u00f6l\u00e7ekleme katsay\u0131s\u0131d\u0131r: \\(|\\det A|\\), d\u00f6n\u00fc\u015f\u00fcm\u00fcn uzunluklar\u0131 (2\u00d72), alanlar\u0131 (2B\/3B ba\u011flamda 2\u00d72 ve 3\u00d73) ya da genel olarak hacimleri ka\u00e7 kat b\u00fcy\u00fctt\u00fc\u011f\u00fcn\u00fc s\u00f6yler. Determinant\u0131n 3 olmas\u0131 d\u00f6n\u00fc\u015f\u00fcm\u00fcn hacmi \u00fc\u00e7e katlad\u0131\u011f\u0131, 0 olmas\u0131 ise d\u00f6n\u00fc\u015f\u00fcm\u00fcn uzay\u0131 daha d\u00fc\u015f\u00fck bir boyuta ezdi\u011fi ve bilgi kaybetti\u011fi anlam\u0131na gelir.<\/p>\n<p>Cebirsel olarak determinant h\u0131zl\u0131 bir tersinirlik testidir. Bir kare matrisin tersi, ancak ve ancak \\(\\det A \\neq 0\\) iken vard\u0131r. \\(\\det A = 0\\) oldu\u011funda matris <em>tekil<\/em> (sing\u00fcler) olarak adland\u0131r\u0131l\u0131r: sat\u0131rlar\u0131 (ve s\u00fctunlar\u0131) do\u011frusal ba\u011f\u0131ml\u0131d\u0131r, \\(Ax = b\\) sistemlar\u0131n\u0131n tek bir \u00e7\u00f6z\u00fcm\u00fc yoktur ve \\(A^{-1}\\) mevcut de\u011fildir. Determinant\u0131n do\u011frusal denklem sistemlerini \u00e7\u00f6zmekten (Cramer kural\u0131) \u00f6zde\u011fer bulmaya ve \u00e7apraz \u00e7arp\u0131m hesaplamaya kadar her yerde kar\u015f\u0131n\u0131za \u00e7\u0131kmas\u0131n\u0131n nedeni budur.<\/p>\n<p>Say\u0131y\u0131 nas\u0131l yorumlayaca\u011f\u0131n\u0131z ba\u011flama ba\u011fl\u0131d\u0131r. Kesin de\u011fer elle kontrol ve teori i\u00e7in \u00f6nemlidir; ondal\u0131k de\u011fer ise tahmin i\u00e7in pratiktir. <em>\u0130\u015faretin<\/em> de anlam\u0131 vard\u0131r: negatif determinant, d\u00f6n\u00fc\u015f\u00fcm\u00fcn bir ayna yans\u0131mas\u0131 gibi y\u00f6n\u00fc ters \u00e7evirdi\u011fini g\u00f6sterir. \u015eunu da unutmay\u0131n: determinant yaln\u0131zca kare matrisler i\u00e7in tan\u0131ml\u0131d\u0131r \u2014 kare olmayan bir matrisin determinant\u0131 yoktur.<\/p>\n<h2>Hesaplay\u0131c\u0131 Nas\u0131l Hesapl\u0131yor: \u0130ki Y\u00f6ntem<\/h2>\n<p>2\u00d72 ve 3\u00d73 matrisler i\u00e7in hesaplay\u0131c\u0131 ilk sat\u0131r boyunca <strong>kofakt\u00f6r a\u00e7\u0131l\u0131m\u0131<\/strong> kullan\u0131r ve her min\u00f6r ile \u00e7arp\u0131m\u0131, 3\u00d73 matrislerde ayr\u0131ca bir Sarrus kural\u0131 \u00e7apraz kontrol\u00fcyle birlikte g\u00f6sterir. 4\u00d74'ten 6\u00d76'ya kadar olan matrislerde ise kesin rasyonel aritmetik kullanan <strong>Gauss eliminasyonuna<\/strong> ge\u00e7er: bir sat\u0131r\u0131n kat\u0131n\u0131 ba\u015fka bir sat\u0131ra eklemek determinant\u0131 de\u011fi\u015ftirmez, buna kar\u015f\u0131l\u0131k her sat\u0131r de\u011fi\u015fimi i\u015fareti ters \u00e7evirir. Determinant, elde edilen \u00fcst \u00fc\u00e7gen matrisin pivotlar\u0131n\u0131n \u00e7arp\u0131m\u0131, yap\u0131lan her de\u011fi\u015fim i\u00e7in \\((-1)\\) \u00e7arp\u0131m\u0131yla beraber bulunur. Bir s\u00fctunda pivot bulunam\u0131yorsa matris tekrilirdir ve determinant tam olarak 0'd\u0131r.<\/p>\n<p>T\u00fcm aritmetik tam say\u0131 ve kesirlerle (kayan noktal\u0131 ondal\u0131klar yerine) yap\u0131ld\u0131\u011f\u0131 i\u00e7in sonu\u00e7 <em>kesindir<\/em> \u2014 en zorlu kesirlerde bile yuvarlama hatas\u0131 olu\u015fmaz. 12 basama\u011fa kadar olan ondal\u0131k giri\u015fler i\u00e7eride tam kesirlere d\u00f6n\u00fc\u015ft\u00fcr\u00fcl\u00fcr; b\u00f6ylece 0,1 yakla\u015f\u0131k bir ikili de\u011fer olarak de\u011fil, tam olarak 1\/10 olarak i\u015flem g\u00f6r\u00fcr.<\/p>\n<h2>\u0130\u015finize Yarayacak \u00d6zellikler ve S\u0131n\u0131rlamalar<\/h2>\n<ul>\n<li>\\(\\det(AB) = \\det A \\cdot \\det B\\); tersinir bir matris i\u00e7in \\(\\det(A^{-1}) = 1\/\\det A\\).<\/li>\n<li>\u0130ki sat\u0131r\u0131n (veya s\u00fctunun) yer de\u011fi\u015ftirmesi determinant\u0131 \u22121 ile \u00e7arpar; bir sat\u0131r\u0131n \\(k\\) sabitiyle \u00e7arp\u0131lmas\u0131 determinant\u0131 \\(k\\) ile \u00e7arpar. \u0130ki \u00f6zde\u015f sat\u0131r\u0131 ya da s\u0131f\u0131r sat\u0131r\u0131 olan bir matrisin determinant\u0131 0'd\u0131r.<\/li>\n<li>\\(\\det A = \\det A^{T}\\) oldu\u011fundan sat\u0131r ve s\u00fctun arg\u00fcmanlar\u0131 birbirinin yerine kullan\u0131labilir. Devri\u011fi (transpozu) kendiniz incelemek isterseniz <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/matris-devrigi-hesaplama\/\">Matris Devri\u011fi (Transpoz) hesaplama arac\u0131n\u0131<\/a> kullanabilirsiniz.<\/li>\n<li>Hesaplay\u0131c\u0131 yaln\u0131zca 2\u00d72 ile 6\u00d76 aras\u0131 boyutlar\u0131, en fazla 12 ondal\u0131k basamakl\u0131 giri\u015fleri kabul eder. Daha b\u00fcy\u00fck matrisler i\u00e7in kofakt\u00f6r a\u00e7\u0131l\u0131ml\u0131 eliminasyon kullan\u0131n ya da b\u00fcy\u00fck do\u011frusal sistemler i\u00e7in tasarlanm\u0131\u015f yaz\u0131l\u0131mlara ge\u00e7in.<\/li>\n<li>Ak\u0131lda kalmas\u0131 gereken pratik bir k\u0131sayol: \\(n \\times n\\) bir matris i\u00e7in \\(\\det(kA) = k^{n} \\det A\\), \u00e7\u00fcnk\u00fc her sat\u0131r \\(k\\) ile \u00e7arp\u0131l\u0131r. <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/matris-skaler-carpim-hesaplama\/\">Matris Skaler \u00c7arp\u0131m hesaplama arac\u0131<\/a> bu t\u00fcr matrisleri do\u011frudan olu\u015fturman\u0131z\u0131 sa\u011flar.<\/li>\n<li>S\u0131f\u0131rdan farkl\u0131 bir determinant, bir do\u011frusal sistemin tek \u00e7\u00f6z\u00fcm\u00fcn\u00fcn oldu\u011funca garanti eder; bu y\u00fczden sistemleri do\u011frudan \u00e7\u00f6zmek isterseniz <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/yok-etme-yontemi\/\">Eliminasyon Y\u00f6ntemi hesaplama arac\u0131n\u0131<\/a> da kullanabilir, ya da k\u00f6\u015fegendeki elemanlar\u0131n toplam\u0131 gibi ilgili de\u011fi\u015fmezleri <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/matris-izi-hesaplama\/\">Matris \u0130zi hesaplama arac\u0131yla<\/a> kar\u015f\u0131la\u015ft\u0131rabilirsiniz.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>2\u00d72'den 6\u00d76'ya kadar herhangi bir kare matrisin determinant\u0131n\u0131 tam olarak hesaplay\u0131n. De\u011ferleri tam say\u0131, ondal\u0131kl\u0131 say\u0131 veya kesir olarak girin; kesin sonu\u00e7, ondal\u0131k de\u011fer, tekil (sing\u00fcler) olup olmad\u0131\u011f\u0131 denetimi ve ad\u0131m ad\u0131m \u00e7\u00f6z\u00fcm\u00fc an\u0131nda al\u0131n.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[54],"class_list":["post-4183","post","type-post","status-publish","format-standard","hentry","category-matematik","tag-matris"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/4183","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/comments?post=4183"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/4183\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/media?parent=4183"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/categories?post=4183"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/tags?post=4183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}