{"id":992,"date":"2024-03-22T11:36:58","date_gmt":"2024-03-22T05:36:58","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=992"},"modified":"2024-10-24T17:32:33","modified_gmt":"2024-10-24T11:32:33","slug":"properties-of-determinant-and-adjoint-matrix","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/properties-of-determinant-and-adjoint-matrix\/","title":{"rendered":"\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09a7\u09b0\u09cd\u09ae \u0993 \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 (Properties Of Determinant &#038; Adjoint Matrix)"},"content":{"rendered":"<h2><span style=\"color: #339966;\"><b>\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09cc\u09b2\u09bf\u0995 \u09a7\u09b0\u09cd\u09ae\u09b8\u09ae\u09c2\u09b9 (Properties of Determinant)<\/b><\/span><\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf \u0995\u09cb\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0995\u09cb\u09a8 \u09b8\u09be\u09b0\u09bf\u09b0 (\u0995\u09b2\u09be\u09ae\u09c7\u09b0) \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09af\u09bc \u09a4\u09ac\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09af\u09bc\u0964 \u09af\u09c7\u09ae\u09a8:\n<p><span class=\"katex-eq\" data-katex-display=\"false\">D=\\begin{vmatrix}\n\na_1 &amp; b_1 &amp; 0\\\\\n\na_2 &amp; b_2 &amp; 0\\\\\n\na_3 &amp; b_3 &amp; 0\n\n\\end{vmatrix}=0<\/span><br \/>\n<\/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 \u098f\u09ac\u0982 \u0995\u09b2\u09be\u09ae \u09b8\u09ae\u09c2\u09b9 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8 \u09ac\u09bf\u09a8\u09bf\u09ae\u09af\u09bc \u0995\u09b0\u09b2\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8\u09c7\u09b0 \u0995\u09cb\u09a8 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u09b9\u09af\u09bc \u09a8\u09be\u0964 \u09af\u09c7\u09ae\u09a8:\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>\u00a0<span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">D'=\\begin{vmatrix}\n\na_1 &amp; a_2 &amp; a_3\\\\\n\nb_1 &amp; b_2 &amp; b_3\\\\\n\nc_1 &amp; c_2 &amp; c_3\n\n\\end{vmatrix}, \\therefore D=D'<\/span><\/p>\n<p><\/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 \u09aa\u09be\u09b6\u09be\u09aa\u09be\u09b6\u09bf \u09a6\u09c1\u0987\u099f\u09bf \u0995\u09b2\u09be\u09ae (\u09b8\u09be\u09b0\u09bf) \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8 \u09ac\u09bf\u09a8\u09bf\u09ae\u09af\u09bc \u0995\u09b0\u09b2\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u099a\u09bf\u09b9\u09cd\u09a8 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09b9\u09af\u09bc \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09ae\u09be\u09a8 \u0985\u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09a5\u09be\u0995\u09c7\u0964<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09ae\u09a8: <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}=-\\begin{vmatrix}\n\nb_1 &amp; a_1 &amp; c_1\\\\\n\nb_2 &amp; a_2 &amp; c_2\\\\\n\nb_3 &amp; a_3 &amp; c_3\n\n\\end{vmatrix}<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <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}=-\\begin{vmatrix}\n\na_2 &amp; b_2 &amp; c_2\\\\\n\na_1 &amp; b_1 &amp; c_1\\\\\n\na_3 &amp; b_3 &amp; c_3\n\n\\end{vmatrix}<\/span><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf \u0995\u09cb\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u0995\u09b2\u09be\u09ae (\u09b8\u09be\u09b0\u09bf) \u0985\u09ad\u09bf\u09a8\u09cd\u09a8 \u09b9\u09af\u09bc \u09a4\u09ac\u09c7, \u09a8\u09bf\u09f0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u09b6\u09c1\u09a8\u09cd\u09af \u09b9\u09ac\u09c7\u0964<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09ae\u09a8: <span class=\"katex-eq\" data-katex-display=\"false\">D=\\begin{vmatrix}\n\na_1 &amp; 1 &amp; 1\\\\\n\na_2 &amp; 1 &amp; 1\\\\\n\na_3 &amp; 1 &amp; 1\n\n\\end{vmatrix}<\/span><\/span>\u00a0<span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">D'=\\begin{vmatrix}\n\na_1 &amp; a_1 &amp; c_1\\\\\n\na_2 &amp; a_2 &amp; c_2\\\\\n\na_3 &amp; a_3 &amp; c_3\n\n\\end{vmatrix}=0<\/span><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09af\u09c7 \u0995\u09cb\u09a8 \u09b8\u09be\u09b0\u09bf (\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb\u0995\u09c7 \u09a4\u09be\u09a6\u09c7\u09b0 \u09a8\u09bf\u099c \u09a8\u09bf\u099c \u09b8\u09b9\u0997\u09c1\u09a3\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09c7 \u0997\u09c1\u09a3\u09ab\u09b2\u0997\u09c1\u09b2\u09cb\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf \u09a8\u09bf\u09b2\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc\u0964\u00a0<\/span><\/li>\n<li><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf \u09a8\u09bf\u09f0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0995\u09cb\u09a8 \u09b8\u09be\u09b0\u09bf (\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb\u0995\u09c7 \u0985\u09aa\u09b0 \u098f\u0995\u099f\u09bf \u09b8\u09be\u09b0\u09bf (\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u0985\u09a8\u09c1\u09b0\u09c2\u09aa \u0989\u09aa\u09be\u09a6\u09be\u09a8\u09c7\u09b0 \u09b8\u09b9\u0997\u09c1\u09a3\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09be \u09b9\u09af\u09bc \u09a4\u09be\u09b9\u09b2\u09c7, \u0997\u09c1\u09a3\u09ab\u09b2\u0997\u09c1\u09b2\u09cb\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/li>\n<\/ul>\n<p><b>\u09af\u09c7\u09ae\u09a8: <\/b><span class=\"katex-eq\" data-katex-display=\"false\">a_2A_1+b_2B_1+c_2C_1=0\\\\\n\na_2B_1+a_2B_2+a_3B_3=0<\/span><span style=\"font-weight: 400;\"> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf&#8230;<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0995\u09cb\u09a8 \u09b8\u09be\u09b0\u09bf (\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u099f\u09bf \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0995\u09c7 \u0995\u09cb\u09a8\u09cb \u09b8\u09cd\u09a5\u09bf\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7, \u09a8\u09bf\u09f0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8\u0995\u09c7\u0993 \u09b8\u09c7\u0987 \u09b8\u09cd\u09a5\u09bf\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a3 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964\u00a0<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09ae\u09a8: <span class=\"katex-eq\" data-katex-display=\"false\">D=k\\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}=\\begin{vmatrix}\n\nka_1 &amp; kb_1 &amp; kc_1\\\\\n\na_2 &amp; b_2 &amp; c_2\\\\\n\na_3 &amp; b_3 &amp; c_3\n\n\\end{vmatrix}=\\begin{vmatrix}\n\nka_1 &amp; b_1 &amp; c_1\\\\\n\nka_2 &amp; b_2 &amp; c_2\\\\\n\nka_3 &amp; b_3 &amp; c_3\n\n\\end{vmatrix}<\/span><\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0995\u09cb\u09a8\u09cb \u09b8\u09be\u09b0\u09bf(\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb \u0985\u09a8\u09cd\u09af \u098f\u0995\u099f\u09bf \u09b8\u09be\u09b0\u09bf(\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u0985\u09a8\u09c1\u09b0\u09c2\u09aa \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb\u09b0 m \u0997\u09c1\u09a3\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u09b9\u09b2\u09c7, \u09a8\u09bf\u09f0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09ac\u09c7\u0964\u00a0<\/span><\/li>\n<li><span style=\"font-weight: 400;\">\u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0995\u09cb\u09a8\u09cb \u09b8\u09be\u09b0\u09bf (\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u0989\u09aa\u09be\u09a6\u09be\u09a8 \u09a6\u09c1\u099f\u09bf \u09aa\u09a6 \u09a8\u09bf\u09af\u09bc\u09c7 \u0997\u09a0\u09bf\u09a4 \u09b9\u09af\u09bc \u09a4\u09be\u09b9\u09b2\u09c7, \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u099f\u09bf \u0985\u09aa\u09b0 \u09a6\u09c1\u0987\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf\u09b0\u09c2\u09aa\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09ac\u09c7\u0964<\/span><\/li>\n<\/ul>\n<p><b>\u09af\u09c7\u09ae\u09a8:\u00a0<\/b><span class=\"katex-eq\" data-katex-display=\"false\">D=k\\begin{vmatrix}\n\na_1+1 &amp; b_1 &amp; c_1\\\\\n\na_2+1 &amp; b_2 &amp; c_2\\\\\n\na_3+1 &amp; b_3 &amp; c_3\n\n\\end{vmatrix}=\\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}=\\begin{vmatrix}\n\n1 &amp; b_1 &amp; c_1\\\\\n\n1 &amp; b_2 &amp; c_2\\\\\n\n1 &amp; b_3 &amp; c_3\n\n\\end{vmatrix}<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0995\u09cb\u09a8 \u09b8\u09be\u09b0\u09bf (\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u0989\u09aa\u09be\u09a6\u09be\u09a8 \u0985\u09a8\u09cd\u09af \u098f\u0995\u099f\u09bf \u09b8\u09be\u09b0\u09bf (\u0995\u09b2\u09be\u09ae)-\u098f\u09b0 \u0985\u09a8\u09c1\u09b0\u09c2\u09aa \u0989\u09aa\u09be\u09a6\u09be\u09a8\u09c7\u09b0 \u098f\u0995\u0987 \u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf \u09ac\u09be \u09b9\u09cd\u09b0\u09be\u09b8 \u0995\u09b0\u09be \u09b9\u09b2\u09c7, \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8\u09c7\u09b0 \u0995\u09cb\u09a8 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u09b9\u09af\u09bc \u09a8\u09be\u0964<\/span><\/li>\n<\/ul>\n<h2><span style=\"color: #339966;\"><b>\u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 (Adjoint matrix)<\/b><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">\u0995\u09cb\u09a8\u09cb \u09ac\u09b0\u09cd\u0997 <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=w5NRpyGBPWA&amp;list=PL0dr4HGr8HPjbTz1tOzTYdUTKm8A49qAf\" target=\"_blank\" rel=\"noopener\">\u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8<\/a><\/span> A \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995 |<\/span><span style=\"font-weight: 400;\">A|<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u09b8\u09b9\u0997\u09c1\u09a3\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09a0\u09bf\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u09c7\u09b0 \u099f\u09cd\u09b0\u09be\u09a8\u09cd\u09b8\u09aa\u09cb\u099c \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u0995\u09c7 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 A \u098f\u09b0 \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 \u09ac\u09b2\u09be \u09b9\u09af\u09bc \u098f\u09ac\u0982 \u098f \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u0995\u09c7 <\/span><span style=\"font-weight: 400;\">Adj\u00a0A<\/span><span style=\"font-weight: 400;\"> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><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><span style=\"font-weight: 400;\">\u098f\u09b0 \u09b8\u09b9\u0997\u09c1\u09a3\u0995 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 <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><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">adj 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}^T=\\begin{bmatrix}\n\nA_{11} &amp; A_{21} &amp; A_{31}\\\\\n\nA_{12} &amp; A_{22} &amp; A_{32}\\\\\n\nA_{13} &amp; A_{23} &amp; A_{33}\n\n\\end{bmatrix}<\/span>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u0996\u09be\u09a8\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">A_{ij}<\/span><\/span><span style=\"font-weight: 400;\"> \u09b9\u099a\u09cd\u099b\u09c7 (i, j) \u09a4\u09ae \u09b8\u09b9\u0997\u09c1\u09a3\u0995 (<span class=\"katex-eq\" data-katex-display=\"false\">a_{ij}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09a4\u09ae \u09ad\u09c1\u0995\u09cd\u09a4\u09bf\u09b0 \u09b8\u09b9\u0997\u09c1\u09a3\u0995)<\/span><\/p>\n<h3><span style=\"color: #800080;\"><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3 : <\/b><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\begin{bmatrix}\n\n1 &amp; -4 &amp; 2\\\\\n\n-2 &amp; 1 &amp; 0\\\\\n\n-1 &amp; -1 &amp; 1\n\n\\end{bmatrix}<\/span><\/span>\u00a0<b>\u098f\u09b0 \u0985\u09a8\u09c1\u09ac\u09a8\u09cd\u09a7\u09c0 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 (Adjoint matrix) \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964<\/b><\/span><\/h3>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\begin{bmatrix}\n\n1 &amp; -4 &amp; 2\\\\\n\n-2 &amp; 1 &amp; 0\\\\\n\n-1 &amp; -1 &amp; 1\n\n\\end{bmatrix}<\/span> \u098f\u09b0 \u09b8\u09b9\u0997\u09c1\u09a3\u0995 \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8 <span class=\"katex-eq\" data-katex-display=\"false\">\\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>\u00a0<span style=\"font-weight: 400;\">\u09b9\u09b2\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">A_{11}=\\begin{vmatrix}\n\n1 &amp; 0\\\\\n\n-1 &amp; 1\n\n\\end{vmatrix}, A_{12}=\\begin{vmatrix}\n\n-2 &amp; 0\\\\\n\n-1 &amp; -1\n\n\\end{vmatrix},A_{13}=\\begin{vmatrix}\n\n-2 &amp; 1\\\\\n\n-1 &amp; -1\n\n\\end{vmatrix},A_{21}=\\begin{vmatrix}\n\n-4 &amp; 2\\\\\n\n-1 &amp; 1\n\n\\end{vmatrix}, A_{22}=\\begin{vmatrix}\n\n1 &amp; 2\\\\\n\n-1 &amp; 1\n\n\\end{vmatrix}, A_{23}=\\begin{vmatrix}\n\n1 &amp; -4\\\\\n\n-1 &amp; -1\n\n\\end{vmatrix}, A_{31}=\\begin{vmatrix}\n\n-4 &amp; 2\\\\\n\n1 &amp; 0\n\n\\end{vmatrix}, A_{32}=\\begin{vmatrix}\n\n1 &amp; 2\\\\\n\n-2 &amp; 0\n\n\\end{vmatrix}, A_{33}=\\begin{vmatrix}\n\n1 &amp; -4\\\\\n\n-2 &amp; 0\n\n\\end{vmatrix}<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">adj A=\\begin{bmatrix}\n\n1+0 &amp; -(-2+0) &amp; 2+1\\\\\n\n-(-4+2) &amp; 1+2 &amp; -(-1-4)\\\\\n\n0-2 &amp; -(0+4) &amp; 1-8\n\n\\end{bmatrix}^T=\\begin{bmatrix}\n\n1 &amp; 2 &amp; 3\\\\\n\n2 &amp; 3 &amp; 5\\\\\n\n-2 &amp; -4 &amp; -7\n\n\\end{bmatrix}=\\begin{bmatrix}\n\n1 &amp; 2 &amp; -2\\\\\n\n2 &amp; 3 &amp; -4\\\\\n\n3 &amp; 5 &amp; -7\n\n\\end{bmatrix}<\/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 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>\u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09cc\u09b2\u09bf\u0995 \u09a7\u09b0\u09cd\u09ae\u09b8\u09ae\u09c2\u09b9 (Properties of Determinant) \u09af\u09a6\u09bf \u0995\u09cb\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u0995\u09cb\u09a8 \u09b8\u09be\u09b0\u09bf\u09b0 (\u0995\u09b2\u09be\u09ae\u09c7\u09b0) \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0997\u09c1\u09b2\u09cb \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09af\u09bc \u09a4\u09ac\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8 \u09b6\u09c2\u09a8\u09cd\u09af \u09b9\u09af\u09bc\u0964 \u09af\u09c7\u09ae\u09a8: \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09b8\u09be\u09b0\u09bf \u098f\u09ac\u0982 \u0995\u09b2\u09be\u09ae \u09b8\u09ae\u09c2\u09b9 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8 \u09ac\u09bf\u09a8\u09bf\u09ae\u09af\u09bc \u0995\u09b0\u09b2\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09ae\u09be\u09a8\u09c7\u09b0 \u0995\u09cb\u09a8 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u09b9\u09af\u09bc \u09a8\u09be\u0964 \u09af\u09c7\u09ae\u09a8: \u00a0\u098f\u09ac\u0982 \u09a8\u09bf\u09b0\u09cd\u09a3\u09be\u09af\u09bc\u0995\u09c7\u09b0 \u09aa\u09be\u09b6\u09be\u09aa\u09be\u09b6\u09bf<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/properties-of-determinant-and-adjoint-matrix\/\">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":[869,870,868,871,873,872],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/992"}],"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=992"}],"version-history":[{"count":7,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/992\/revisions"}],"predecessor-version":[{"id":16084,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/992\/revisions\/16084"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=992"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=992"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=992"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}