{"id":991,"date":"2024-03-22T11:37:45","date_gmt":"2024-03-22T05:37:45","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=991"},"modified":"2024-10-24T17:02:31","modified_gmt":"2024-10-24T11:02:31","slug":"determinant","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/determinant\/","title":{"rendered":"\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 (Determinant)"},"content":{"rendered":"<p><span style=\"font-weight: 400;\">\u09e7\u09ec\u09ef\u09e9 \u0996\u09cd\u09b0\u09bf\u09b8\u09cd\u099f\u09be\u09ac\u09cd\u09a6\u09c7 \u0997\u09a3\u09bf\u09a4\u09ac\u09bf\u09a6 \u09b2\u09bf\u09ac\u09a8\u09bf\u099c (Leibnitz) \u09b8\u09b0\u09b2 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09ae\u09be\u09b2\u09be\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u09c7 \u098f\u0995 \u09ac\u09bf\u09b6\u09c7\u09b7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09c7\u09b0 \u0985\u09ac\u09a4\u09be\u09b0\u09a3\u09be \u0995\u09b0\u09c7\u09a8\u0964 \u0989\u09a8\u09ac\u09bf\u0982\u09b6 \u09b6\u09a4\u09be\u09ac\u09cd\u09a6\u09c0\u09a4\u09c7 \u0997\u09be\u0989\u09b8 (Gauss) \u098f\u09ac\u0982 \u0995\u09b8\u09bf (Cauchy) \u098f \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09c7 \u0986\u09b0\u0993 \u09b8\u09c1\u09b8\u09cd\u09aa\u09b7\u09cd\u099f \u09a7\u09be\u09b0\u09a3\u09be \u09a6\u09c7\u09a8\u0964 \u09b8\u09b0\u09cd\u09ac\u09aa\u09cd\u09b0\u09a5\u09ae \u0995\u09b8\u09bf \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09a8\u09be\u09ae \u09a6\u09c7\u09a8 <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=w5NRpyGBPWA&amp;list=PL0dr4HGr8HPjbTz1tOzTYdUTKm8A49qAf\" target=\"_blank\" rel=\"noopener\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995<\/a><\/span> (Determinant).<\/span><\/p>\n<h2><b><span style=\"color: #339966;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 (Determinant):<\/span> <\/b><\/h2>\n<p><span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09b9\u09b2\u09cb \u09ac\u09bf\u09b6\u09c7\u09b7 \u0986\u0995\u09be\u09b0\u09c7 \u09b2\u09bf\u0996\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u098f\u0995 \u09aa\u09cd\u09b0\u0995\u09be\u09b0\u09c7\u09b0 \u09b0\u09be\u09b6\u09bf\u0964 \u0995\u09cb\u09a8\u09cb \u09ac\u09b0\u09cd\u0997 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{bmatrix}\n\na_1 &amp; a_2\\\\\n\nb_1 &amp; b_2\n\n\\end{bmatrix}<\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb\u0995\u09c7 \u098f\u0995\u0987 \u09b0\u09c7\u0996\u09c7 \u098f\u09ac\u0982 \u09a4\u09be\u09a6\u09c7\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u09a8\u09be \u0995\u09b0\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{vmatrix}\n\na_1 &amp; a_2\\\\\n\nb_1 &amp; b_2\n\n\\end{vmatrix}<\/span><\/span>\u00a0<span style=\"font-weight: 400;\">\u0986\u0995\u09be\u09b0\u09c7 \u09b2\u09bf\u0996\u09b2\u09c7 \u098f\u0995\u09c7 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09ac\u09b0\u09cd\u0997 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09ac\u09be \u09b8\u0982\u0995\u09cd\u09b7\u09c7\u09aa\u09c7 \u09b6\u09c1\u09a7\u09c1 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09be\u09a8\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">a_1, a_2, b_1, b_2<\/span><\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8 \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">a_1, a_2<\/span><\/span><span style=\"font-weight: 400;\"> \u09b9\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u099f\u09bf\u09b0 \u09aa\u09cd\u09b0\u09a7\u09be\u09a8 \u0995\u09b0\u09cd\u09a3 \u0997\u09a0\u09a8\u0995\u09be\u09b0\u09c0 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0964<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb\u09b0 \u0985\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09ac\u09bf\u09a8\u09cd\u09af\u09be\u09b8\u0995\u09c7 \u09b8\u09be\u09b0\u09bf (row) \u098f\u09ac\u0982 \u0989\u09b2\u09ae\u09cd\u09ac \u09ac\u09bf\u09a8\u09cd\u09af\u09be\u09b8\u0995\u09c7 \u09b8\u09cd\u09a4\u09ae\u09cd\u09ad \u09ac\u09be (column) \u09ac\u09b2\u09c7\u0964\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09b8\u09be\u09b0\u09bf \u0993 \u0995\u09b2\u09be\u09ae \u09b8\u0982\u0996\u09cd\u09af\u09be \u0985\u09ac\u09b6\u09cd\u09af\u0987 \u09b8\u09ae\u09be\u09a8 \u09b9\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a4\u09cd\u09b0\u09be: \u0995\u09cb\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09b8\u09be\u09b0\u09bf \u0993 \u0995\u09b2\u09be\u09ae \u09b8\u0982\u0996\u09cd\u09af\u09be n \u09b9\u09b2\u09c7, \u09a4\u09be\u0995\u09c7 n \u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/li>\n<\/ul>\n<h2><span style=\"color: #339966;\"><b>\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09aa\u09a6 :<\/b><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">\u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{vmatrix}\n\na_1 &amp; b_1 &amp; c_1\\\\\n\na_2 &amp; b_2 &amp; c_2\\\\\n\na_3 &amp; b_3 &amp; c_3\n\n\\end{vmatrix}<\/span><\/span>\u00a0<span style=\"font-weight: 400;\">\u098f\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\">a_1, b_1, c_1<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0997\u09c1\u09a3\u09ab\u09b2\u0995\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09aa\u09a6 (terms) \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964\u00a0<\/span><\/p>\n<h3><b><span style=\"color: #800080;\">\u09ae\u09c2\u0996\u09cd\u09af \u0995\u09b0\u09cd\u09a3 \u0993 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09bf\u0995 \u0995\u09b0\u09cd\u09a3:<\/span> <\/b><\/h3>\n<p><span style=\"font-weight: 400;\">\u0989\u09aa\u09b0\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09b2\u0995\u09cd\u09b7\u09cd\u09af \u0995\u09b0\u09b2\u09c7 \u09a6\u09c7\u0996\u09be \u09af\u09be\u09af\u09bc <span class=\"katex-eq\" data-katex-display=\"false\">a_1, b_2 \u0993 c_3<\/span><\/span><span style=\"font-weight: 400;\"> \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb \u098f\u0995\u099f\u09bf \u0995\u09b0\u09cd\u09a3 \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">a_3, b_2, c_1<\/span><\/span><span style=\"font-weight: 400;\"> \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb \u0985\u09aa\u09b0 \u098f\u0995\u099f\u09bf \u0995\u09b0\u09cd\u09a3 \u0997\u09a0\u09a8 \u0995\u09b0\u09c7\u0964 \u09aa\u09cd\u09b0\u09a5\u09ae \u0995\u09b0\u09cd\u09a3\u0995\u09c7 <strong>\u09ae\u09c1\u0996\u09cd\u09af \u0995\u09b0\u09cd\u09a3<\/strong> \u098f\u09ac\u0982 \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc \u0995\u09b0\u09cd\u09a3\u0995\u09c7 <strong>\u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09bf\u0995 \u0995\u09b0\u09cd\u09a3<\/strong> \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964 \u09ae\u09c2\u0996\u09cd\u09af\u0995\u09b0\u09cd\u09a3\u09c7\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2 <span class=\"katex-eq\" data-katex-display=\"false\">a_1, b_2 , c_3<\/span><\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 <strong>\u09ae\u09c1\u0996\u09cd\u09af\u09aa\u09a6<\/strong> \u098f\u09ac\u0982 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09bf\u0995 \u0995\u09b0\u09cd\u09a3\u09c7\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2\u0995\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">a_3, b_2, c_1<\/span><\/span><span style=\"font-weight: 400;\">\u00a0<strong>\u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09bf\u0995 \u09aa\u09a6<\/strong> \u09ac\u09b2\u09c7\u0964<\/span><\/p>\n<p><img loading=\"lazy\" class=\"aligncenter\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2018\/11\/maths\/2015\/12\/24070546\/Determinant-Of-A-Matrix.jpg\" alt=\"\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \" width=\"800\" height=\"400\" \/><\/p>\n<h2><span style=\"color: #339966;\"><b>\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc (Determining of Determinant of Matrix)<\/b><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">\u098f\u0995 \u0995\u09cd\u09f0\u09ae\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8: <span class=\"katex-eq\" data-katex-display=\"false\">A=[a_{11}]<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995<\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\">|A|=|a_{11}|=a_{11}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0995\u09cb\u09a8\u09cb \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09be\u09a4\u09cd\u09b0 \u09b8\u09be\u09b0\u09bf \u0993 \u098f\u0995\u099f\u09bf \u09ae\u09be\u09a4\u09cd\u09b0 \u0995\u09b2\u09be\u09ae \u09a5\u09be\u0995\u09b2\u09c7 \u098f\u09b0 \u09ae\u09be\u09a8 \u09b9\u09ac\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u099f\u09bf \u09af\u09c7 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u0990 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0987\u0964\u00a0<\/span><\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3: <\/b>|-6|<span style=\"font-weight: 400;\">=6\u00a0<\/span><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0, |<\/span><span style=\"font-weight: 400;\">5|<\/span><span style=\"font-weight: 400;\">=5<\/span><\/p>\n<p><span style=\"font-weight: 400;\">2\u00d72 \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8: <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{bmatrix}\n\na_{11} &amp; a_{12}\\\\\n\na_{21} &amp; a_{22}\n\n\\end{bmatrix}<\/span><\/span>\u00a0<span style=\"font-weight: 400;\">\u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 <span class=\"katex-eq\" data-katex-display=\"false\">|A|=a_{11}.a_{22}-a_{12}.a_{21}<\/span><\/span><\/p>\n<h3><span style=\"color: #800080;\"><strong>3\u00d73<\/strong><\/span><b><span style=\"color: #800080;\"> \u0986\u0995\u09be\u09b0\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 :<\/span> <\/b><\/h3>\n<p><span style=\"font-weight: 400;\">\u09a4\u09bf\u09a8\u099f\u09bf \u09b8\u09be\u09b0\u09bf \u0993 \u09a4\u09bf\u09a8\u099f\u09bf \u0995\u09b2\u09be\u09ae \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09be\u09b0 \u09b8\u09b9\u099c\u09c7 \u0995\u09b0\u09be\u09b0 \u099c\u09a8\u09cd\u09af \u099c\u09a8\u09cd\u09af <em>\u09b8\u09be\u09b0\u09b8 \u09a1\u09be\u09df\u09be\u0997\u09cd\u09b0\u09be\u09ae<\/em> (Sarrus Diagram) \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09be \u09b9\u09df\u0964 \u098f \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u0997\u09c1\u09b2\u09bf\u0995\u09c7 \u09aa\u09be\u09b6\u09be\u09aa\u09be\u09b6\u09bf \u09a6\u09c1\u0987\u09ac\u09be\u09b0 \u09b2\u09bf\u0996\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09ae\u09cd\u09a8\u09cb\u0995\u09cd\u09a4\u09ad\u09be\u09ac\u09c7 \u09ac\u09c7\u09b0 \u0995\u09b0\u09be \u09b9\u09df\u0964 <\/span><b>\u00a0<\/b><\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">A=\\begin{bmatrix}\n\na_{11} &amp; a_{12} &amp; a_{13}\\\\\n\na_{21} &amp; a_{22} &amp; a_{23}\\\\\n\na_{31} &amp; a_{32} &amp; a_{33}\n\n\\end{bmatrix}<\/span> <span style=\"font-weight: 400;\">\u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u00a0\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">|A|=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{11}a_{23}a_{32}-a_{12}a_{23}a_{31}-a_{13}a_{22}a_{31}......(i)\\\\\n\n= 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\n=a_{11}\\begin{vmatrix}\n\na_{22} &amp; a_{23}\\\\\n\na_{32} &amp; a_{33}\n\n\\end{vmatrix}-a_{12}\\begin{vmatrix}\n\na_{21} &amp; a_{23}\\\\\n\na_{31} &amp; a_{32}\n\n\\end{vmatrix}+a_{13}\\begin{vmatrix}\n\na_{21} &amp; a_{22}\\\\\n\na_{31} &amp; a_{32}\n\n\\end{vmatrix}......(ii)<\/span>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a8\u09cd\u09af\u09ad\u09be\u09ac\u09c7 <\/span><span style=\"font-weight: 400;\">(ii)<\/span><span style=\"font-weight: 400;\"> \u09b9\u09a4\u09c7 \u09ac\u09b2\u09be \u09af\u09be\u09df <\/span><span style=\"font-weight: 400;\">{(<\/span><span style=\"font-weight: 400;\">1,1)<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">\u09a4\u09ae \u09ad\u09c1\u0995\u09cd\u09a4\u09bf <\/span><span style=\"font-weight: 400;\">\u00d7(<\/span><span style=\"font-weight: 400;\">\u09e7\u09ae \u09b8\u09be\u09b0\u09bf \u0993 \u09e7\u09ae \u0995\u09b2\u09be\u09ae \u09ac\u09be\u09a6\u09c7 \u0985\u09ac\u09b6\u09bf\u09b7\u09cd\u099f \u0985\u0982\u09b6\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995<\/span><span style=\"font-weight: 400;\">)}-{(<\/span><span style=\"font-weight: 400;\">1,2)<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">\u09a4\u09ae \u09ad\u09c1\u0995\u09cd\u09a4\u09bf<\/span><span style=\"font-weight: 400;\">\u00a0\u00d7(<\/span><span style=\"font-weight: 400;\">\u09e7\u09ae \u09b8\u09be\u09b0\u09bf \u0993 \u09e8\u09df \u0995\u09b2\u09be\u09ae \u09ac\u09be\u09a6\u09c7 \u0985\u09ac\u09b6\u09bf\u09b7\u09cd\u099f \u0985\u0982\u09b6\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995<\/span><span style=\"font-weight: 400;\">)}+{(<\/span><span style=\"font-weight: 400;\">1,3)<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">\u09a4\u09ae \u09ad\u09c1\u0995\u09cd\u09a4\u09bf<\/span><span style=\"font-weight: 400;\">\u00a0\u00d7(<\/span><span style=\"font-weight: 400;\">\u09e7\u09ae \u09b8\u09be\u09b0\u09bf \u0993 \u09e9\u09df \u0995\u09b2\u09be\u09ae \u09ac\u09be\u09a6\u09c7 \u0985\u09ac\u09b6\u09bf\u09b7\u09cd\u099f \u0985\u0982\u09b6\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995<\/span><span style=\"font-weight: 400;\">)}\u00a0<\/span><span style=\"font-weight: 400;\">\u09b8\u09b0\u09b2 \u0995\u09b0\u09b2\u09c7 \u09a4\u09bf\u09a8 \u09a7\u09b0\u09a8\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df\u0964\u00a0<\/span><\/p>\n<h3><b>\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af:<\/b><\/h3>\n<ul>\n<li><span style=\"font-weight: 400;\"> \u0989\u09b2\u09cd\u09b2\u09bf\u0996\u09bf\u09a4 \u09a8\u09bf\u09df\u09ae \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b8\u09be\u09b0\u09bf \u09ac\u09be \u0995\u09b2\u09be\u09ae \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09c3\u09a4\u09bf \u0995\u09b0\u09b2\u09c7\u0993 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u098f\u0995\u0987 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09ac\u09c7\u0964 <\/span><\/li>\n<li>\u098f\u0996\u09be\u09a8\u09c7 \u09b6\u09c1\u09a7\u09c1\u09ae\u09be\u09a4\u09cd\u09b0 \u09aa\u09cd\u09b0\u09a5\u09ae \u09b8\u09be\u09b0\u09bf \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09c3\u09a4\u09bf \u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09b9\u09df\u09c7\u099b\u09c7, \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b8\u09be\u09b0\u09bf \u09ac\u09be \u0995\u09b2\u09be\u09ae \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09c3\u09a4\u09bf \u0995\u09b0\u09be\u09b0 \u09a8\u09bf\u09df\u09ae \u09a8\u09bf\u09ae\u09cd\u09a8\u09b0\u09c2\u09aa : \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995 = \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09be\u09b0\u09bf \u09ac\u09be \u0995\u09b2\u09be\u09ae\u09c7\u09b0 {(\u09e7\u09ae \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09ad\u09c1\u0995\u09cd\u09a4\u09bf) (\u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u099f\u09bf \u09a7\u09be\u09b0\u09a3\u0995\u09be\u09b0\u09c0 \u09b8\u09be\u09b0\u09bf \u0993 \u0995\u09b2\u09be\u09ae \u09ac\u09cd\u09af\u09a4\u09c0\u09a4 \u0985\u09ac\u09b6\u09bf\u09b7\u09cd\u099f \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995)} + {(\u09e8\u09df \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09ad\u09c1\u0995\u09cd\u09a4\u09bf) (\u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u099f\u09bf \u09a7\u09be\u09b0\u09a3\u0995\u09be\u09b0\u09c0 \u09b8\u09be\u09b0\u09bf \u0993 \u0995\u09b2\u09be\u09ae \u09ac\u09cd\u09af\u09a4\u09c0\u09a4 \u0985\u09ac\u09b6\u09bf\u09b7\u09cd\u099f \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995)}+ {(\u09e9\u09df \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09ad\u09c1\u0995\u09cd\u09a4\u09bf) (\u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u099f\u09bf \u09a7\u09be\u09b0\u09a3\u0995\u09be\u09b0\u09c0 \u09b8\u09be\u09b0\u09bf \u0993 \u0995\u09b2\u09be\u09ae \u09ac\u09cd\u09af\u09a4\u09c0\u09a4 \u0985\u09ac\u09b6\u09bf\u09b7\u09cd\u099f \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995)}<\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">(p,q)<\/span><b>&#8211; \u09a4\u09ae \u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 <\/b><strong>=<span class=\"katex-eq\" data-katex-display=\"false\">(-1)^{p+q}<\/span><\/strong><\/p>\n<p><span style=\"font-weight: 400;\">\u0989\u09aa\u09b0\u09bf-\u0989\u0995\u09cd\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u099f\u09bf\u0995\u09c7 \u09a4\u09c3\u09a4\u09c0\u09df \u09b8\u09be\u09b0\u09bf \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09c3\u09a4\u09bf \u0995\u09b0\u09b2\u09c7,\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{vmatrix}\n\n2 &amp; 1 &amp; 1\\\\\n\n0 &amp; 5 &amp; -2\\\\\n\n1 &amp; -3 &amp; 4\n\n\\end{vmatrix}=(-1)^{3+1}(1)\\begin{vmatrix}\n\n1 &amp; 1\\\\\n\n5 &amp; -2\n\n\\end{vmatrix}+(-1)^{3+2}(-3)\\begin{vmatrix}\n\n2 &amp; 1\\\\\n\n0 &amp; -2\n\n\\end{vmatrix}+(-1)^{3+3}(4)\\begin{vmatrix}\n\n2 &amp; 1\\\\\n\n0 &amp; 5\n\n\\end{vmatrix}\\\\\n\n=(-2-5)+3(-4)+4\\times 10+-7-12+40=21<\/span>\n<p><span style=\"font-weight: 400;\">\u0986\u09ac\u09be\u09b0 \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09df \u0995\u09b2\u09be\u09ae \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09c3\u09a4\u09bf \u0995\u09b0\u09b2\u09c7,\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{vmatrix}\n\n2 &amp; 1 &amp; 1\\\\\n\n0 &amp; 5 &amp; -2\\\\\n\n1 &amp; -3 &amp; 4\n\n\\end{vmatrix}=(-1)^{1+2}(1)\\begin{vmatrix}\n\n0 &amp; -2\\\\\n\n1 &amp; 4\n\n\\end{vmatrix}+(-1)^{2+2}(5)\\begin{vmatrix}\n\n2 &amp; 1\\\\\n\n1 &amp; 4\n\n\\end{vmatrix}+(-1)^{3+2}(-3)\\begin{vmatrix}\n\n2 &amp; 1\\\\\n\n0 &amp; -2\n\n\\end{vmatrix}\\\\\n\n=-(2)+5(8-1)+3(-4)=-2+35-12=21<\/span>\n<p>&nbsp;<\/p>\n<h2><span style=\"color: #339966;\"><b>\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u0985\u09a8\u09c1\u09b0\u09be\u09b6\u09bf \u098f\u09ac\u0982 \u09b8\u09b9\u0997\u09c1\u09a3\u0995 (Minor and Cofactor of Determinant)<\/b><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8\u09cb \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995 D \u098f\u09b0 \u0995\u09cb\u09a8\u09cb \u09ad\u09c1\u0995\u09cd\u09a4\u09bf \u09af\u09c7 \u09b8\u09be\u09b0\u09bf \u0993 \u09af\u09c7 \u0995\u09b2\u09be\u09ae\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b8\u09c7 \u09b8\u09be\u09b0\u09bf \u0993 \u09b8\u09c7 \u0995\u09b2\u09be\u09ae \u09ac\u09cd\u09af\u09a4\u09c0\u09a4 \u0985\u09ac\u09b6\u09bf\u09b7\u09cd\u099f \u09ad\u09c1\u0995\u09cd\u09a4\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u0995\u09c7 \u0989\u0995\u09cd\u09a4 \u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u09b0 \u0985\u09a8\u09c1\u09b0\u09be\u09b6\u09bf \u09ac\u09b2\u09be \u09b9\u09df\u0964\u00a0<\/span><\/p>\n<p><b>\u09af\u09c7\u09ae\u09a8: <\/b><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{vmatrix}\n\na_{11} &amp; a_{12} &amp; a_{13}\\\\\n\na_{21} &amp; a_{22} &amp; a_{23}\\\\\n\na_{31} &amp; a_{32} &amp; a_{33}\n\n\\end{vmatrix}<\/span>\u00a0<span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 <span class=\"katex-eq\" data-katex-display=\"false\">a_{11},a_{12},a_{13}<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u0985\u09b0\u09cd\u09a5\u09be\u09ce (<\/span><span style=\"font-weight: 400;\">1,1)<\/span><span style=\"font-weight: 400;\">, (<\/span><span style=\"font-weight: 400;\">1,2)<\/span><span style=\"font-weight: 400;\">, (<\/span><span style=\"font-weight: 400;\">1,3)<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">\u09a4\u09ae \u0985\u09a8\u09c1\u09b0\u09be\u09b6\u09bf <\/span><span style=\"font-weight: 400;\">\u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{vmatrix}\n\na_{22} &amp; a_{23}\\\\\n\na_{32} &amp; a_{33}\n\n\\end{vmatrix}, \\begin{vmatrix}\n\na_{21} &amp; a_{23}\\\\\n\na_{31} &amp; a_{33}\n\n\\end{vmatrix}, \\begin{vmatrix}\n\na_{21} &amp; a_{22}\\\\\n\na_{31} &amp; a_{32}\n\n\\end{vmatrix}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09a5\u09be\u09af\u09a5 \u099a\u09bf\u09b9\u09cd\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u0995\u09cb\u09a8\u09cb \u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u09b0 \u0985\u09a8\u09c1\u09b0\u09be\u09b6\u09bf\u0995\u09c7 \u09b8\u09c7 \u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u09b0 \u09b8\u09b9\u0997\u09c1\u09a3\u0995 \u09ac\u09b2\u09be \u09b9\u09df\u0964 \u0995\u09cb\u09a8\u09cb \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 <\/span><span style=\"font-weight: 400;\">(r,c)<\/span><span style=\"font-weight: 400;\">\u2013\u09a4\u09ae \u09b8\u09b9\u0997\u09c1\u09a3\u0995\u09c7\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 \u09b9\u09ac\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">(-1)^{r+c}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u0985\u09b0\u09cd\u09a5\u09be\u09ce <\/span><span style=\"font-weight: 400;\">(r+c)<\/span><span style=\"font-weight: 400;\"> \u099c\u09cb\u09dc \u09b9\u09b2\u09c7 \u2018+\u2019 \u099a\u09bf\u09b9\u09cd\u09a8 \u098f\u09ac\u0982 \u09ac\u09bf\u099c\u09cb\u09dc \u09b9\u09b2\u09c7 \u2018\u2212\u2019 \u099a\u09bf\u09b9\u09cd\u09a8 \u09b9\u09ac\u09c7\u0964 \u09a4\u09c3\u09a4\u09c0\u09df \u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09df\u0995\u09c7\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{vmatrix}\n\n+ &amp; - &amp; +\\\\\n\n- &amp; + &amp; -\\\\\n\n+ &amp; - &amp; +\n\n\\end{vmatrix}<\/span><\/span>\u00a0<span style=\"font-weight: 400;\">\u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0995\u09cb\u09a3\u09be\u0995\u09cb\u09a3\u09bf \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u0989\u09aa\u09be\u09a6\u09be\u09a8 \u09eb\u099f\u09bf \u099a\u09bf\u09b9\u09cd\u09a8 \u2018+\u2019 \u098f\u09ac\u0982 \u0985\u09aa\u09b0 \u09ea\u099f\u09bf\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 \u2018\u2212\u2019\u00a0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<h3><span style=\"color: #800080;\"><b>\u09b8\u09b9\u0997\u09c1\u09a3\u0995\u09c7\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 \u09b8\u09a8\u09be\u0995\u09cd\u09a4\u0995\u09b0\u09a3:<\/b><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09af\u09c7 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u09c7\u09b0 \u09b8\u09b9\u0997\u09c1\u09a3\u0995 \u09ac\u09c7\u09b0 \u0995\u09b0\u09a4\u09c7 \u09b9\u09b2\u09c7 \u0989\u0995\u09cd\u09a4 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u099f\u09bf \u09af\u09a4 \u09a8\u0982 \u0995\u09b2\u09be\u09ae \u0993 \u09af\u09a4 \u09a8\u0982 \u09b8\u09be\u09b0\u09bf\u09a4\u09c7 \u0986\u099b\u09c7 \u09a4\u09be\u09a6\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u099c\u09cb\u09a1\u09bc \u09b8\u0982\u0996\u09cd\u09af\u09be \u09b9\u09b2\u09c7, \u09b8\u09b9\u0997\u09c1\u09a3\u0995\u09c7\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 <\/span><span style=\"font-weight: 400;\">(+ve)<\/span><span style=\"font-weight: 400;\"> \u09b9\u09ac\u09c7 \u098f\u09ac\u0982 \u09af\u09cb\u0997\u09ab\u09b2 \u09ac\u09bf\u099c\u09cb\u09a1\u09bc \u09b9\u09b2\u09c7, \u099a\u09bf\u09b9\u09cd\u09a8 \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 <\/span><span style=\"font-weight: 400;\">(-ve)<\/span><span style=\"font-weight: 400;\"> \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/p>\n<p><b>\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09c3\u09a4\u09bf<\/b><span style=\"font-weight: 400;\"> : <\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09c3\u09a4\u09bf \u09ac\u09c1\u099d\u09be\u09a8\u09cb\u09b0 \u099c\u09a8\u09cd\u09af \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u0995\u09cd\u09b0\u09ae\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09ac\u09bf\u09ac\u09c7\u099a\u09a8\u09be \u0995\u09b0\u09bf\u0964<\/span><\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">D=\\begin{vmatrix}\n\na_1 &amp; b_1 &amp; c_1\\\\\n\na_2 &amp; b_2 &amp; c_2\\\\\n\na_3 &amp; b_3 &amp; c_3\n\n\\end{vmatrix}<\/span> <span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u099f\u09bf\u0995\u09c7 \u09aa\u09cd\u09b0\u09a5\u09ae \u0995\u09b2\u09be\u09ae \u09ac\u09b0\u09be\u09ac\u09b0 \u09ac\u09bf\u09b8\u09cd\u09a4\u09be\u09b0 \u0995\u09b0\u09bf\u0964 \u098f \u099c\u09a8\u09cd\u09af <span class=\"katex-eq\" data-katex-display=\"false\">a_1<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09ac\u09b0\u09be\u09ac\u09b0 \u0995\u09b2\u09be\u09ae \u0993 \u09b8\u09be\u09b0\u09bf \u09ac\u09be\u09a6 \u09a6\u09bf\u09b2\u09c7 \u09ac\u09be\u0995\u09bf \u099a\u09be\u09b0\u099f\u09bf \u0989\u09aa\u09be\u09a6\u09be\u09a8 \u09a8\u09bf\u09af\u09bc\u09c7 \u098f\u0995\u099f\u09bf <\/span><span style=\"font-weight: 400;\">2\u00d72<\/span><span style=\"font-weight: 400;\"> \u0995\u09cd\u09f0\u09ae\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09b9\u09af\u09bc, \u09af\u09be <span class=\"katex-eq\" data-katex-display=\"false\">a_1<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u0997\u09c1\u09a3 \u0995\u09b0\u09bf\u0964 \u098f\u09b0\u09aa\u09b0 \u09aa\u09cd\u09b0\u09a5\u09ae \u0995\u09b2\u09be\u09ae \u098f\u09b0 \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc \u0989\u09aa\u09be\u09a6\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\">a_2<\/span><\/span><span style=\"font-weight: 400;\"> \u09a8\u09bf\u09af\u09bc\u09c7 \u09aa\u09c2\u09b0\u09cd\u09ac\u09c7\u09b0 \u09a8\u09bf\u09af\u09bc\u09ae\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">a_2<\/span><\/span><span style=\"font-weight: 400;\"> \u09ac\u09b0\u09be\u09ac\u09b0 \u0995\u09b2\u09be\u09ae \u0993 \u09b8\u09be\u09b0\u09bf \u09ac\u09be\u09a6 \u09a6\u09bf\u09b2\u09c7 \u09ac\u09be\u0995\u09bf \u099a\u09be\u09b0\u099f\u09bf \u0989\u09aa\u09be\u09a6\u09a8 \u09a6\u09bf\u09af\u09bc\u09c7 \u098f\u0995\u099f\u09bf <\/span><span style=\"font-weight: 400;\">2\u00d72<\/span><span style=\"font-weight: 400;\"> \u0995\u09cd\u09f0\u09ae\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09b9\u09af\u09bc, \u09af\u09be\u09b0 \u09aa\u09c2\u09b0\u09cd\u09ac\u09c7 \u098f\u0995\u099f\u09be \u098b\u09a3\u09be\u09a4\u09cd\u09ae\u0995 \u099a\u09bf\u09b9\u09cd\u09a8 \u09a6\u09bf\u09af\u09bc\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">a_2<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 \u0997\u09c1\u09a3 \u0995\u09b0\u09bf \u098f\u09ac\u0982 \u09b8\u09ac \u09b6\u09c7\u09b7\u09c7 \u09aa\u09cd\u09b0\u09a5\u09ae \u0995\u09b2\u09be\u09ae\u09c7\u09b0 \u09b6\u09c7\u09b7 \u09ac\u09be \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u0989\u09aa\u09be\u09a6\u09a8 <span class=\"katex-eq\" data-katex-display=\"false\">a_3<\/span><\/span><span style=\"font-weight: 400;\"> \u09a8\u09bf\u09af\u09bc\u09c7 \u09aa\u09c2\u09b0\u09cd\u09ac\u09c7\u09b0 \u09a8\u09bf\u09af\u09bc\u09ae\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 <\/span><span style=\"font-weight: 400;\">2\u00d72<\/span><span style=\"font-weight: 400;\"> \u0995\u09cd\u09b0\u09ae\u09c7\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 <\/span><span style=\"font-weight: 400;\">a<\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\"> \u0997\u09c1\u09a3 \u0995\u09b0\u09bf\u0964 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u09ae\u09be\u09a8\u0997\u09c1\u09b2\u09bf\u0987 \u09b9\u099a\u09cd\u099b\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u099f\u09bf\u09b0 \u099a\u09c2\u09a1\u09bc\u09be\u09a8\u09cd\u09a4 \u09ac\u09bf\u09b8\u09cd\u09a4\u09c3\u09a4\u09bf\u0964 \u09a8\u09bf\u099a\u09c7 \u09a4\u09be \u09a6\u09c7\u0996\u09be\u09a8 \u09b9\u09b2:\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">D=a_1\\begin{vmatrix}\n\nb_2 &amp; b_3\\\\\n\nc_2 &amp; c_3\n\n\\end{vmatrix}-a_2\\begin{vmatrix}\n\nb_1 &amp; c_1\\\\\n\nb_3 &amp; c_3\n\n\\end{vmatrix}+a_3\\begin{vmatrix}\n\nb_1 &amp; c_1\\\\\n\nb_2 &amp; c_2\n\n\\end{vmatrix}\\\\\n\n=a_1(b_2c_3-b_3c_2)-a_2(c_3b_1-b_3c_1)+a_3(b_1c_2-b_2c_1)\\\\\n\n=(a_1b_2c_3+b_1c_2a_3+c_1a_2b_3)-(a_3b_2c_1+b_3c_2a_1+c_3a_2b_1)<\/span>\n<p>&nbsp;<\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-25-online-batch-2-bangla-english-ict\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a \u09e8.\u09e6 (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-26-online-batch-bangla-english-ict\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2025-online-batch\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2026-online-batch\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/medical-admission-course\/\">\u09ae\u09c7\u09a1\u09bf\u0995\u09c7\u09b2 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-a-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-b-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/buet-ques-solve\/\">\u09ac\u09c1\u09df\u09c7\u099f \u0995\u09cb\u09b6\u09cd\u099a\u09c7\u09a8 \u09b8\u09b2\u09ad \u0995\u09cb\u09b0\u09cd\u09b8<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-a-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-b-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u09b8\u09cd\u0995\u09bf\u09b2 \u09a1\u09c7\u09ad\u09c7\u09b2\u09aa\u09ae\u09c7\u09a8\u09cd\u099f \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/study-abroad-complete-guideline\/\">\u09ac\u09bf\u09a6\u09c7\u09b6\u09c7 \u0989\u099a\u09cd\u099a\u09b6\u09bf\u0995\u09cd\u09b7\u09be: Study Abroad Complete Guideline<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/student-hacks\/\">Student Hacks<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ielts-course\/\">IELTS Course by Munzereen Shahid<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/english-grammar-course\/\">Complete English Grammar Course<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ms-bundle\/\"> Microsoft Office 3 in 1 Bundle<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ghore-boshe-freelancing\/\">\u0998\u09b0\u09c7 \u09ac\u09b8\u09c7 Freelancing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/facebook-marketing\/\">Facebook Marketing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/adobe-4-in-1-bundle\/\">Adobe 4 in 1 Bundle<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><em>\u09e7<\/em><em>\u09e6 \u09ae\u09bf\u09a8\u09bf\u099f \u09b8\u09cd\u0995\u09c1\u09b2\u09c7\u09b0 \u0995\u09cd\u09b2\u09be\u09b8\u0997\u09c1\u09b2\u09cb \u0985\u09a8\u09c1\u09b8\u09b0\u09a3 \u0995\u09b0\u09a4\u09c7 \u09ad\u09bf\u099c\u09bf\u099f: <span style=\"color: #993300;\"><strong><a style=\"color: #993300;\" href=\"https:\/\/10minuteschool.com\/?ref=https%3A%2F%2Fblog.10minuteschool.com%2Fwordpress%2F&amp;post_id=78178&amp;blog_category_id=700\">www.10minuteschool.com<\/a><\/strong><\/span><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u09e7\u09ec\u09ef\u09e9 \u0996\u09cd\u09b0\u09bf\u09b8\u09cd\u099f\u09be\u09ac\u09cd\u09a6\u09c7 \u0997\u09a3\u09bf\u09a4\u09ac\u09bf\u09a6 \u09b2\u09bf\u09ac\u09a8\u09bf\u099c (Leibnitz) \u09b8\u09b0\u09b2 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09ae\u09be\u09b2\u09be\u09b0 \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u09c7 \u098f\u0995 \u09ac\u09bf\u09b6\u09c7\u09b7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09c7\u09b0 \u0985\u09ac\u09a4\u09be\u09b0\u09a3\u09be \u0995\u09b0\u09c7\u09a8\u0964 \u0989\u09a8\u09ac\u09bf\u0982\u09b6 \u09b6\u09a4\u09be\u09ac\u09cd\u09a6\u09c0\u09a4\u09c7 \u0997\u09be\u0989\u09b8 (Gauss) \u098f\u09ac\u0982 \u0995\u09b8\u09bf (Cauchy) \u098f \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09c7 \u0986\u09b0\u0993 \u09b8\u09c1\u09b8\u09cd\u09aa\u09b7\u09cd\u099f \u09a7\u09be\u09b0\u09a3\u09be \u09a6\u09c7\u09a8\u0964 \u09b8\u09b0\u09cd\u09ac\u09aa\u09cd\u09b0\u09a5\u09ae \u0995\u09b8\u09bf \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09a8\u09be\u09ae \u09a6\u09c7\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 (Determinant). \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 (Determinant): \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 \u09b9\u09b2\u09cb \u09ac\u09bf\u09b6\u09c7\u09b7 \u0986\u0995\u09be\u09b0\u09c7<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/determinant\/\">Read More<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4256,3037,50,3026],"tags":[867,868,609,611],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/991"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=991"}],"version-history":[{"count":7,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/991\/revisions"}],"predecessor-version":[{"id":16083,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/991\/revisions\/16083"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=991"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=991"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=991"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}