{"id":5643,"date":"2020-10-21T10:50:58","date_gmt":"2020-10-21T01:50:58","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=5643"},"modified":"2021-06-03T23:17:58","modified_gmt":"2021-06-03T14:17:58","slug":"linear-algebra-determinants","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/linear-algebra-determinants\/","title":{"rendered":"\ud589\ub82c\uc2dd\uacfc \uc5ed\ud589\ub82c"},"content":{"rendered":"<p>\uacc4\uc218\uc640 \uc0c1\uc218\uac00 \uc2e4\uc218\uc778 \uc774\ucc28\ubc29\uc815\uc2dd\uc774 \uc2e4\uc218 \ubc94\uc704\uc5d0\uc11c \uba87 \uac1c\uc758 \ud574\ub97c \uac16\ub294\uc9c0 \uc54c\uc544\ubcf4\uae30 \uc704\ud574\uc11c\ub294 \ud310\ubcc4\uc2dd\uc758 \ubd80\ud638\ub97c \uc0b4\ud3b4\ubcf4\uba74 \ub41c\ub2e4. \uc774\uc640 \ube44\uc2b7\ud558\uac8c \uc815\uc0ac\uac01\ud589\ub82c\uc758 \uc5ed\ud589\ub82c\uc774 \uc874\uc7ac\ud558\ub294\uc9c0 \uc54c\uc544\ubcf4\ub294 \uacf5\uc2dd\uc774 \uc788\ub294\ub370, \uadf8\uac83\uc774 \ud589\ub82c\uc2dd\uc774\ub2e4. \ud589\ub82c\uc2dd\uc740 \ud2b9\uc815\ud55c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc120\ud615\ubc94\ud568\uc218\ub85c \uc815\uc758\ub420 \uc218\ub3c4 \uc788\ub294\ub370, \uadf8\ub7ec\ud55c \ud568\uc218\ub294 \ud06c\uae30\uac00 \uc791\uc740 \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \ud06c\uae30\uac00 \ud070 \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc744 \uacc4\uc0b0\ud558\ub294 \uadc0\ub0a9\uc801\uc778 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \ub610\ud55c \ud589\ub82c\uc2dd\uc740 \ud589\ub82c\uc758 \uac01 \uc131\ubd84\ub4e4\uc744 \uc774\uc6a9\ud558\uc5ec \uc9c1\uc811 \uacc4\uc0b0\ud558\ub294 \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ub420 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud589\ub82c\uc2dd\uc758 \ub450 \uac00\uc9c0 \uc815\uc758\ub97c \uc0b4\ud3b4\ubcf4\uace0 \ub450 \uc815\uc758\uac00 \uc11c\ub85c \ub3d9\uce58\uc784\uc744 \ubc1d\ud78c\ub2e4. \ub354\ubd88\uc5b4 \ud589\ub82c\uc758 \uac00\uc5ed\uc131\uacfc \uad00\ub828\ub41c \ud589\ub82c\uc2dd\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<div style=\"display: none; visibility: hidden;\">\n\\[<br \/>\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}<br \/>\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}<br \/>\n\\newcommand{\\proj}{{\\operatorname{proj}}}<br \/>\n\\newcommand{\\adj}{{\\operatorname{adj}}}<br \/>\n\\]\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\uc120\ud615\ubc94\ud568\uc218\ub85c\uc11c\uc758 \ud589\ub82c\uc2dd<\/h2>\n<p>\ud589\ub82c\uc2dd\uc744 \uc815\uc758\ud558\uae30 \uc704\ud574 \uba87 \uac00\uc9c0 \ud45c\uae30\ubc95\uc744 \uc815\ud558\uc790. \\(K\\)\uac00 \uccb4\uc774\uace0 \\(A\\in M_n (K),\\) \\(n > 1\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(1\\le i \\le n ,\\) \\(1\\le j \\le n\\)\uc778 \uc790\uc5f0\uc218 \\(i,\\) \\(j\\)\uc5d0 \ub300\ud558\uc5ec \ud589\ub82c<br \/>\n\\[\\partial_{ij} A \\in M_{n-1}(K)\\]<br \/>\n\ub97c \\(A\\)\uc758 \\(i\\)\uc9f8 \ud589\uacfc \\(j\\)\uc9f8 \uc5f4\uc744 \uc81c\uac70\ud55c \ud589\ub82c\ub85c \uc815\uc758\ud558\uc790. \uc608\ucee8\ub300<br \/>\n\\[\\partial_{12}<br \/>\n\\left(\\begin{array}{ccc} a_{11} &#038; a_{12} &#038; a_{13} \\\\ a_{21} &#038; a_{22} &#038; a_{23} \\\\ a_{31} &#038; a_{32} &#038; a_{33} \\end{array}\\right)<br \/>\n=<br \/>\n\\left(\\begin{array}{cc} a_{21} &#038; a_{23} \\\\ a_{31} &#038; a_{33} \\end{array}\\right)\\]<br \/>\n\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \ud589\ub82c \\(A\\)\ub97c \uc5f4\ubca1\ud130\ub4e4\uc744 \ubd99\uc5ec\uc11c<br \/>\n\\[A = (A^1 ,\\, A^2 ,\\, \\cdots ,\\, A^n )\\]<br \/>\n\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uae30\uc5b5\ud558\uc790. \uc5ec\uae30\uc11c \\(A^j\\)\ub294 \\(A\\)\uc758 \\(j\\)\uc9f8 \uc5f4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc5f4\ubca1\ud130\uc774\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p class=\"marginbottom0\"><span class=\"theorem\">\uc815\ub9ac 1. (\ud589\ub82c\uc2dd\uc758 \uae30\ubcf8\uc815\ub9ac)<\/span><\/p>\n<p>\\(n\\ge 1\\)\uc77c \ub54c, \uc138 \uc870\uac74 (i), (ii), (iii)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218<br \/>\n\\[\\det : M_n (K) \\rightarrow K\\]<br \/>\n\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis-roman\">\n<li><span class=\"defined\">\ub2e4\uc911\uc120\ud615\uc131\uc9c8<\/span>(multilinearity).&nbsp; \\(A = (A^1 ,\\, \\cdots ,\\, A^n ) \\in M_n (K)\\)\uc774\uace0, \\(A\\)\uc758 \\(j\\)\uc9f8 \uc5f4\ubca1\ud130\uac00 \\[A^j = \\lambda_1 C_1 + \\lambda_2 C_2\\]\ub85c \ub098\ud0c0\ub09c\ub2e4\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(C_1 ,\\, C_2 \\in K^n\\)\uc774\uace0 \\(\\lambda_1 ,\\) \\(\\lambda_2\\)\ub294 \uc2a4\uce7c\ub77c\uc774\ub2e4. \uadf8\ub7ec\uba74<br \/>\n\\[\\det(A) = \\lambda_1 \\det (A^1 ,\\, \\cdots ,\\, C_1 ,\\, \\cdots ,\\, A^n ) + \\lambda_2 ( A^1 ,\\, \\cdots ,\\, C_2 ,\\, \\cdots ,\\, A^n )\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c \uc6b0\ubcc0\uc758 \\(C_1\\)\uacfc \\(C_2\\)\ub294 \uac01 \ud589\ub82c\uc758 \\(j\\)\uc9f8 \uc5f4\uc774\ub2e4. \uc989 \\(\\det\\)\ub294 \ud589\ub82c\uc758 \uac01 \uc5f4\uc5d0\uc11c \uc120\ud615\uc774\ub2e4.\n<\/li>\n<li><span class=\"defined\">\ubd80\ud638\uad50\ub300\uc131\uc9c8<\/span>(alternation of sign).&nbsp; \\(A = (A^1 ,\\, \\cdots ,\\ A^n)\\)\uc774\uace0, \uc801\ub2f9\ud55c \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(A^j = A^{j+1}\\)\uc774\ub77c\uace0 \ud558\uc790. \uc989 \\(A\\)\uc758 \uc5f4 \uc911\uc5d0\uc11c \uc11c\ub85c \uc778\uc811\ud558\uace0 \ub3d9\uc77c\ud55c \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\det(A) =0\\)\uc774\ub2e4.<\/li>\n<li><span class=\"defined\">\uc815\uaddc\ud654\uc131\uc9c8<\/span>(normalization).&nbsp; \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\det(I_n)=1\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\uc774\ub7ec\ud55c \ud568\uc218 \\(\\det\\)\ub97c <span class=\"defined\">\ud589\ub82c\uc2dd<\/span>(determinant)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uc5ec\uae30\uc11c\ub294 \uc138 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(\\det\\)\uac00 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc790. \uadf8\ub7ec\ud55c \ud568\uc218\uac00 \uc720\uc77c\ud558\ub2e4\ub294 \uc0ac\uc2e4\uc740 \ub4a4\uc5d0\uc11c \uc99d\uba85\ud560 \uac83\uc774\ub2e4.<\/p>\n<p>\\(n=1\\)\uc778 \uacbd\uc6b0 \\(A=(a)\\)\uc774\ubbc0\ub85c \ub2f9\uc5f0\ud788 \\(\\det(A) = a\\)\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \\(n > 1\\)\uc774\ub77c\uace0 \ud558\uace0, \\(\\det : M_n (K) \\rightarrow K\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[\\det(A) = \\sum_{j=1}^n (-1)^{j+1} a_{1j} \\det(\\partial_{1j} A).\\tag{1}\\]<br \/>\n\uc774\ub807\uac8c \uc815\uc758\ub41c \ud568\uc218 \\(\\det\\)\uac00 (i), (ii), (iii)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc790.<\/p>\n<p style=\"text-align: left;\"><span class=\"definition\">(i)\uc758 \uc99d\uba85.<\/span> \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(A^1 = \\lambda_1 C_1 + \\lambda_2 C_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(a_{11} = \\lambda_1 c_1 + \\lambda_2 c_2\\)\uc774\uba70, \\(c_1\\)\uacfc \\(c_2\\)\uac00 \uac01\uac01 \\(C_1\\)\uacfc \\(C_2\\)\uc758 \uccab\uc9f8 \uc131\ubd84\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 (1)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\det(A) = (\\lambda_1 c_1 + \\lambda_2 c_2) \\det(\\partial_{11} A) + \\sum_{j=2}^n (-1)^{j+1} a_{1j} \\det(\\partial_{1j} A)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uadc0\ub0a9\uc801 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\det(A)<br \/>\n&#038;= \\lambda_1 c_1 \\det(\\partial_{11} A) + \\lambda_2 c_2 \\det(\\partial_{11} A) \\\\[5pt]<br \/>\n&#038;\\quad+\\sum_{j=2}^n (-1)^{j+1} a_{1j} \\lambda_1 \\det (\\partial_{1j} (C_1 ,\\, A^2 ,\\, \\cdots ,\\, A^n )) \\\\[5pt]<br \/>\n&#038;\\quad+\\sum_{j=2}^n (-1)^{j+1} a_{1j} \\lambda_2 \\det (\\partial_{1j} (C_1 ,\\, A^2 ,\\, \\cdots ,\\, A^n )). \\tag{2}<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7f0\ub370<br \/>\n\\[\\partial_{11} A = \\partial_{11} (C_1 ,\\, A^2 ,\\, \\cdots ,\\, A^n ) = \\partial_{11}(C_2 ,\\, A^2 ,\\,\\cdots ,\\, A^n )\\]<br \/>\n\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc774 \ub4f1\uc2dd\uc5d0 \uc788\ub294 \uc138 \ud589\ub82c\uc911 \uc5b4\ub290 \uac83\ub3c4 \\(A\\)\uc758 \uccab\uc9f8 \uc5f4\uc744 \ud3ec\ud568\uace0 \uc788\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \ub4f1\uc2dd\uacfc (2)\ub97c \uacb0\ud569\ud558\uc5ec \ub2e4\uc74c \uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\det(A)<br \/>\n&#038;= \\lambda_1 \\left\\{<br \/>\nc_1 \\det(\\partial_{11} (C_1 ,\\, A^2 ,\\, \\cdots ,\\, A^n)) + \\sum_{j=2}^n (-1)^{j+1} a_{1j} \\det(\\partial_{1j} (C_1 ,\\, A^2 ,\\, \\cdots ,\\, A^n))<br \/>\n\\right\\} \\\\[5pt]<br \/>\n&#038;\\quad+ \\lambda_2 \\left\\{<br \/>\nc_2 \\det(\\partial_{11} (C_2 ,\\, A^2 ,\\, \\cdots ,\\, A^n)) + \\sum_{j=2}^n (-1)^{j+1} a_{1j} \\det(\\partial_{1j} (C_2 ,\\, A^2 ,\\, \\cdots ,\\, A^n))<br \/>\n\\right\\}.\\tag{3}<br \/>\n\\end{align}\\]<br \/>\n\uc5ec\uae30\uc11c \\(c_1 \\det(\\partial_{11} (C_1 ,\\, A^2 ,\\, \\cdots ,\\, A^n ))\\)\uc740 \\(\\det(C_1 ,\\, A^2 ,\\, \\cdots ,\\, A^n )\\)\uc744 \uc804\uac1c\ud588\uc744 \ub54c \uccab\uc9f8 \ud56d\uc774\uba70, \\(c_2 \\det(\\partial_{11}(C_2 ,\\, A^2 ,\\, \\cdots ,\\, A^n ))\\)\uc740 \\(\\det(C_2 ,\\, A^2 ,\\, \\cdots ,\\, A^n )\\)\uc744 \uc804\uac1c\ud588\uc744 \ub54c \uccab\uc9f8 \ud56d\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (3)\uc744 \uac04\ub2e8\ud558\uac8c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[\\det(A) = \\lambda_1 \\det(C,\\,A^2 ,\\,\\cdots,\\,A^n) + \\lambda_2 \\det(C_2 ,\\, A^2 ,\\, \\cdots ,\\, A^n ).\\]<br \/>\n\uc774 \uc2dd\uc774 \uace7 (i)\uc758 \uc2dd\uc774\ub2e4.<\/p>\n<p><span class=\"definition\">(ii)\uc758 \uc99d\uba85.<\/span><br \/>\n\uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(A^1 = A^2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 (1)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\det(A) = a_{11}\\det(\\partial_{11}A) &#8211; a_{12}\\det(\\partial_{12}A) + \\sum_{j=3}^n (-1)^{j+1} a_{1j} \\det(\\partial_{1j} A)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(a_{11} = a_{12}\\)\uc774\uace0 \\(\\partial_{11}A = \\partial_{12}A\\)\uc774\ubbc0\ub85c, \uc704 \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc758 \ucc98\uc74c \ub450 \ud56d\uc740 \uc11c\ub85c \uc0c1\uc1c4\ub41c\ub2e4. \uc6b0\ubcc0\uc5d0\uc11c \ub0a8\uc740 \ud56d\uc5d0 \\(\\partial_{1j} A,\\) \\(j=3,\\,4,\\,\\cdots,\\,n\\)\uc774 \uc788\ub294\ub370, \uc774 \ud589\ub82c\uc740 \uccab\uc9f8 \uc5f4\uacfc \ub458\uc9f8 \uc5f4\uc774 \ub3d9\uc77c\ud558\ubbc0\ub85c \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc0c1\uc1c4\ub41c\ub2e4. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \uacc4\uc18d \uc0c1\uc1c4\uc2dc\ud0a4\uba74 \uacb0\uad6d \ub450 \uc5f4\uc774 \uac19\uc740 \\(2\\times 2\\) \ud589\ub82c \ud589\ub82c\uc2dd\ub9cc \ub0a8\ub294\ub370, \uc774\ub7ec\ud55c \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc740 \\(0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\det(A)=0\\)\uc774\ub2e4.<\/p>\n<p><span class=\"definition\">(iii)\uc758 \uc99d\uba85.<\/span><br \/>\n\\(I_n\\)\uc758 \uccab\uc9f8 \ud589\uc5d0\uc11c \\((1,\\,1)\\)-\uc131\ubd84\ub9cc \\(1\\)\uc774\uace0 \ub2e4\ub978 \uc131\ubd84\uc740 \ubaa8\ub450 \\(0\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\det(I_n) = 1\\cdot \\det(\\partial_{11} I_n ) = \\det (I_{n-1})\\]<br \/>\n\uc774\ub2e4. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uba74<br \/>\n\\[\\det(I_n) = \\cdots = \\det(I_1) = \\det(1) = 1\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\\(2\\times 2\\) \ud589\ub82c\uacfc \\(3\\times 3\\) \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\det\\left(\\begin{array}{cc} a &#038; b \\\\ c &#038; d \\end{array}\\right) = ad-bc,\\]<br \/>\n\\[\\det\\left(\\begin{array}{ccc} a_{11} &#038; a_{12} &#038; a_{13} \\\\ a_{21} &#038; a_{22} &#038; a_{23} \\\\ a_{31} &#038; a_{32} &#038; a_{33} \\end{array}\\right)<br \/>\n=<br \/>\na_{11} \\det\\left(\\begin{array}{cc} a_{22} &#038; a_{23} \\\\ a_{32} &#038; a_{33} \\end{array}\\right) &#8211;<br \/>\na_{12} \\det\\left(\\begin{array}{cc} a_{21} &#038; a_{23} \\\\ a_{31} &#038; a_{33} \\end{array}\\right) +<br \/>\na_{13} \\det\\left(\\begin{array}{cc} a_{21} &#038; a_{22} \\\\ a_{31} &#038; a_{32} \\end{array}\\right) .<br \/>\n\\]\n<\/p>\n<\/div>\n<div class=\"corollary margintop2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 2.<\/span><br \/>\n\\(n \\ge 2\\)\uc774\uace0 \\(A\\)\uac00 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis-roman marginbottom0\">\n<li>\\(A\\)\uc758 \uc778\uc811\ud55c \ub450 \uc5f4\uc744 \ubc14\uafb8\uc5b4 \ub9cc\ub4e0 \ud589\ub82c\uc744 \\(A &#8216; \\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\det(A &#8216; ) = -\\det(A)\\)\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \ub450 \uc5f4\uc744 \ubc14\uafb8\uc5b4 \ub9cc\ub4e0 \ud589\ub82c\uc744 \\(A &#8216; \\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\det(A &#8216; ) = -\\det(A)\\)\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc5f4 \uc911\uc5d0\uc11c \uc11c\ub85c \uc77c\uce58\ud558\ub294 \uac83\uc774 \uc788\uc73c\uba74 \\(\\det(A)=0\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p style=\"text-align: left;\"><span class=\"definition\">(i)\uc758 \uc99d\uba85.<\/span><br \/>\n\\(A = (A^1 ,\\, \\cdots ,\\, A^j ,\\, A^{j+1} ,\\, \\cdots ,\\, A^n ),\\) \\(A &#8216; = (A^1 ,\\, \\cdots ,\\, A^{j+1} ,\\, A^j ,\\, \\cdots ,\\, A^n )\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \ub2e4\uc74c\uacfc \uac19\uc740 \ud589\ub82c\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[A &#8216; &#8216; = (A^1 ,\\, \\cdots ,\\, A^j + A^{j+1} ,\\, A^j + A^{j+1} ,\\, \\cdots ,\\, A^n ).\\]<br \/>\n\\(A &#8216; &#8216; \\)\uc740 \uc11c\ub85c \uc77c\uce58\ud558\ub294 \uc778\uc811\ud55c \ub450 \uc5f4\uc744 \uac00\uc9c0\uace0 \uc788\uc73c\ubbc0\ub85c \\(\\det(A &#8216; &#8216; )=0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\det(A) + \\det(A &#8216; )<br \/>\n&#038;= 0 + \\det(A^1 ,\\, \\cdots ,\\, A^j ,\\, A^{j+1} ,\\, \\cdots ,\\, A^n ) \\\\[5pt]<br \/>\n&#038;\\quad + \\det(A^1 ,\\, \\cdots ,\\, A^{j+1} ,\\, A^j ,\\, \\cdots ,\\, A^n ) +0 \\\\[5pt]<br \/>\n&#038;= \\det(A^1 ,\\, \\cdots ,\\, A^j ,\\, A^j ,\\, \\cdots ,\\,A^n ) \\\\[5pt]<br \/>\n&#038;\\quad + \\det(A^1 ,\\, \\cdots ,\\, A^j ,\\, A^{j+1} ,\\, \\cdots ,\\, A^n ) \\\\[5pt]<br \/>\n&#038;\\quad + \\det(A^1 ,\\, \\cdots ,\\, A^{j+1} ,\\, A^j ,\\, \\cdots ,\\, A^n ) \\\\[5pt]<br \/>\n&#038;\\quad + \\det(A^1 ,\\, \\cdots ,\\, A^{j+1} ,\\, A^{j+1} ,\\, \\cdots ,\\, A^n ) \\\\[5pt]<br \/>\n&#038;= \\det(A &#8216; &#8216; ) =0 .<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(\\det(A &#8216; ) = -\\det(A)\\)\uc774\ub2e4.<\/p>\n<p><span class=\"definition\">(ii)\uc758 \uc99d\uba85.<\/span><br \/>\n\ub450 \uc5f4\uc744 \uc11c\ub85c \ubc14\uafb8\ub294 \uc5f0\uc0b0\uc740 \uc778\uc811\ud55c \ub450 \uc5f4\uc744 \uc11c\ub85c \ubc14\uafb8\ub294 \uc5f0\uc0b0\uc744 \ud640\uc218 \ubc88 \ud55c \uac83\uc774\ub2e4. \uc778\uc811\ud55c \uc5f4\uc744 \uc11c\ub85c \ubc14\uafc0 \ub54c\ub9c8\ub2e4 \ud589\ub82c\uc2dd\uc758 \ud06c\uae30\ub294 \uac19\uace0 \ubd80\ud638\ub294 \ubc18\ub300\uac00 \ub418\ubbc0\ub85c, \ub450 \uc5f4\uc744 \uc11c\ub85c \ubc14\uafbc \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc740 \ucc98\uc74c \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc5d0\uc11c \ubd80\ud638\ub9cc \ubc14\uafbc \uac83\uacfc \uac19\ub2e4.<\/p>\n<p><span class=\"definition\">(iii)\uc758 \uc99d\uba85.<\/span><br \/>\n\ub9cc\uc57d \\(A\\)\uc758 \uc778\uc811\ud55c \ub450 \uc5f4\uc774 \uc11c\ub85c \uc77c\uce58\ud558\uba74 \\(A\\)\uc758 \ud589\ub82c\uc2dd\uc740 \ub2f9\uc5f0\ud788 \\(0\\)\uc774\ub2e4. \ub9cc\uc57d \\(A\\)\uc758 \uc11c\ub85c \uac19\uc740 \ub450 \uc5f4\uc774 \uc11c\ub85c \uc778\uc811\ud558\uc9c0 \uc54a\ub2e4\uba74, \uc5f4\uc758 \uc704\uce58\ub97c \uc11c\ub85c \ubc14\uafb8\ub294 \uc5f0\uc0b0\uc744 \ud55c \ubc88 \ud568\uc73c\ub85c\uc368 \uc778\uc811\ud55c \ub450 \uc5f4\uc774 \uc11c\ub85c \uac19\uc740 \ud589\ub82c \\(A &#8216; \\)\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\det(A) = -\\det(A &#8216; ) = 0\\]<br \/>\n\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"corollary margintop2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 3.<\/span><br \/>\n\\(A\\)\uac00 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uace0 \\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc774 \uc77c\ucc28\uc885\uc18d\uc774\uba74 \\(\\det(A) =0\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(A=(0)\\)\uc774\uba74 \ub2f9\uc5f0\ud788 \\(\\det(A)=0\\)\uc774\ub2e4. \uc774\uc81c \\(n \\ge 2\\)\uc774\uace0 \\(A\\)\uac00 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uba70 \\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc774 \uc77c\ucc28\uc885\uc18d\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc2a4\uce7c\ub77c \\(\\lambda_k\\)\ub4e4\uc774 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[A^j = \\sum_{k\\ne j} \\lambda_k A^k\\]<br \/>\n\ub85c \ud45c\ud604\ub41c\ub2e4. \uadf8\ub7ec\uba74<br \/>\n\\[\\det(A) = \\sum_{k\\ne j} \\lambda_k \\det(A^1 ,\\, \\cdots ,\\, A^k ,\\, \\cdots ,\\, A^n )\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(A^k\\)\ub294 \\(j\\)\uc9f8 \uc5f4\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[(A^1 ,\\, \\cdots ,\\, A^k ,\\, \\cdots ,\\, A^n )\\]<br \/>\n\uc740 \uc77c\uce58\ud558\ub294 \ub450 \uc5f4\uc744 \uac00\uc9c4 \ud589\ub82c\uc774\ubbc0\ub85c \ub530\ub984\uc815\ub9ac 2\uc758 (iii)\uc5d0 \uc758\ud558\uc5ec \uc774 \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc740 \\(0\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\ud589\ub82c\uc2dd\uc758 \ub2e4\ub978 \uc815\uc758<\/h2>\n<p>(1)\uc758 \ud589\ub82c\uc2dd\uc740 \ub354 \uc791\uc740 \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \ub354 \ud070 \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc744 \uad6c\ud558\ub294 \uadc0\ub0a9\uc801 \uc815\uc758\uc774\ub2e4. \uc774\uc81c \ud589\ub82c\uc758 \uc131\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \uc9c1\uc811 \ud589\ub82c\uc744 \uacc4\uc0b0\ud558\ub294 \uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc5ec\uae30\uc11c \uc0b4\ud3b4\ubcf4\ub294 \uacf5\uc2dd\uc740 \ud589\ub82c\uc2dd\uc744 \uc2e4\uc81c\ub85c \uacc4\uc0b0\ud558\uae30 \uc704\ud55c \uac83\uc774 \uc544\ub2c8\ub77c \ud589\ub82c\uc2dd\uc758 \uc131\uc9c8\uc744 \ubc1d\ud788\uae30 \uc704\ud55c \uae30\ubc18\uc73c\ub85c\uc11c\uc758 \uc5ed\ud560\uc774 \ub354 \ud06c\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 4.<\/span><br \/>\n\ud589\ub82c \\(A\\in M_n (K)\\)\uc758 \ud589\ub82c\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\det(A) = \\sum_{\\pi\\in S_n} \\sigma(\\pi) a_{\\pi(1)1} \\cdots a_{\\pi(n)n} .\\tag{4}\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\pi\\)\ub294 \ub300\uce6d\uad70 \\(S_n\\)\uc758 \uc6d0\uc18c\uc774\uba70 \\(\\sigma\\)\ub294 \ubd80\ud638 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. [\uc989 \\(\\sigma(\\pi)\\)\ub294 \\(\\pi\\)\uac00 \uc9dd\uc218 \uac1c\uc758 \ud638\ud658\uc758 \ud569\uc131\uc77c \ub54c \\(1,\\) \ud640\uc218 \uac1c\uc758 \ud638\ud658\uc758 \ud569\uc131\uc77c \ub54c \\(-1\\)\uc744 \uac12\uc73c\ub85c \uac16\ub294 \ud568\uc218\uc774\ub2e4.]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(E_j\\)\uac00 \\(n\\times 1\\) \ud589\ub82c\uc774\uace0 \\(j\\)\uc9f8 \uc131\ubd84\uc774 \\(1\\)\uc774\uba70 \ub2e4\ub978 \uc131\ubd84\uc740 \ubaa8\ub450 \\(0\\)\uc778 \ud589\ub82c\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[A = (a_{11} E_1 + \\cdots + a_{n1} E_n ,\\, A^2 ,\\, \\cdots ,\\, A^n )\\]<br \/>\n\uc774\ub2e4. \ud589\ub82c\uc2dd (1)\uc758 \uc120\ud615\uc131\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\det(A) = \\sum_{i=1}^n \\det(E_i ,\\, A^2 ,\\, \\cdots ,\\, A^n )\\]<br \/>\n\uc774\ub2e4. \uac19\uc740 \ub17c\uc99d \uacfc\uc815\uc744 \ub450 \ubc88\uc9f8 \uc5f4\uc5d0 \uc801\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\det(A) = \\sum_{i=1}^n \\sum_{j=1}^n a_{i1} a_{j2} \\det(E_i ,\\, E_j ,\\, \\cdots ,\\, A^n ).\\]<br \/>\n\uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(n\\)\uac1c\uc758 \uc5f4\uc5d0 \uc774 \uacfc\uc815\uc744 \ucc28\ub840\ub85c \uc801\uc6a9\ud558\uba74 \uc5f4\ubca1\ud130 \\(E_j\\)\uc640 \uadf8 \uacc4\uc218\ub97c \uacb0\ud569\ud558\ub294 \ubaa8\ub4e0 \uacbd\uc6b0\ub97c \uc5bb\ub294\ub2e4. \uc815\uc758\uc5ed\uacfc \uacf5\uc5ed\uc774 \ubaa8\ub450 \\(\\left\\{ 1,\\,2,\\,\\cdots,\\,n\\right\\}\\)\uc778 \ubaa8\ub4e0 \ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc744 \\(S\\)\ub77c\uace0 \ud558\uba74, \uc774 \uc0c1\ud669\uc740 \uc2dd\uc73c\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[\\det(A) = \\sum_{\\varphi\\in S} a_{\\varphi (1)1} \\cdots a_{\\varphi(n) n} \\det(E_{\\varphi (1)},\\, \\cdots, \\, E_{\\varphi (n)} ).\\tag{5}\\]<br \/>\n\ub9cc\uc57d \\(i \\ne j\\)\uc774\uba74\uc11c \\(\\varphi(i) = \\varphi(j)\\)\uc778 \\(i,\\) \\(j\\)\uac00 \uc874\uc7ac\ud55c\ub2e4\uba74, \ub530\ub984\uc815\ub9ac 2-(iii)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\det(E_{\\varphi (1)} ,\\, \\cdots ,\\, E_{\\varphi(n)})=0\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(S\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c \\(\\varphi\\)\uc5d0 \ub300\ud558\uc5ec (5)\uc758 \ud569\uc744 \uad6c\ud560 \ud544\uc694\uac00 \uc5c6\uace0 \\(\\varphi\\)\uac00 \uc77c\ub300\uc77c\ub300\uc751\uc77c \ub54c\ub9cc \uad6c\ud558\uba74 \ub41c\ub2e4. \uc989 (5)\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\det(A) = \\sum_{\\pi\\in S_n} a_{\\pi (1)1} \\cdots a_{\\pi(n) n} \\det(E_{\\pi (1)},\\, \\cdots, \\, E_{\\pi (n)} ).\\tag{6}\\]<br \/>\n\uc2dd (6)\uc5d0 \uc788\ub294 \ud589\ub82c\uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790. \\(I_n = (E_1 ,\\, \\cdots ,\\, E_n)\\)\uc758 \uc5f4\uc758 \uc21c\uc11c\ub97c \uc801\ub2f9\ud788 \ubc14\uafb8\uc5b4 \\((E_{\\pi(1)} ,\\, \\cdots ,\\, E_{\\pi(n)})\\)\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \uc5f4\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\ub294 \ud69f\uc218\uc758 \ud640\uc9dd\uc131\uc740 \\(\\pi\\)\uc758 \ubd80\ud638\uc640 \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\det(E_{\\pi (1)} ,\\, \\cdots ,\\, E_{\\pi (n)}) = \\sigma(\\pi) \\det(E_1 ,\\, \\cdots ,\\, E_n ) = \\sigma(\\pi) \\det(I_n) = \\sigma(\\pi)\\tag{7}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. (7)\uacfc (6)\uc744 \uacb0\ud569\ud558\uba74 (4)\ub97c \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"corollary margintop2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 5.<\/span><br \/>\n\uc784\uc758\uc758 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c \\(A\\)\uc5d0 \ub300\ud558\uc5ec \\(\\det(A) = \\det( \\,^t \\! A )\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\ub300\uce6d\uad70\uacfc \uce58\ud658\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\\(\\left\\{ \\pi \\,\\vert\\, \\pi \\in S_n \\right\\}\\)\uacfc \\(\\left\\{ \\pi^{-1} \\,\\vert\\, \\pi \\in S_n \\right\\}\\)\uc740 \uac19\uc740 \uc9d1\ud569\uc774\uba70, \ub450 \uc9d1\ud569\uc758 \uc6d0\uc18c\ub294 \uc77c\ub300\uc77c \ub300\uc751\ub41c\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(\\pi \\in S_n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\sigma(\\pi) = \\sigma(\\pi^{-1})\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \uc0ac\uc2e4\uc744 \uc5fc\ub450\uc5d0 \ub450\uace0 \\(\\det(\\,^t\\!A)\\)\ub97c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\det(\\,^t\\! A)<br \/>\n&#038;= \\sum_{\\pi\\in S_n} \\sigma(\\pi) a_{1\\pi (1)} \\cdots a_{n\\pi (n)} \\\\[5pt]<br \/>\n&#038;= \\sum_{\\pi\\in S_n} \\sigma(\\pi^{-1}) a_{\\pi^{-1} (1)1} \\cdots a_{\\pi^{-1} (n)n} \\\\[5pt]<br \/>\n&#038;= \\sum_{\\pi\\in S_n} \\sigma(\\pi) a_{\\pi (1)1} \\cdots a_{\\pi(n)n} \\\\[5pt]<br \/>\n&#038;= \\det(A).\\tag*{\\(\\blacksquare\\)}<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p>\uc774\uc81c \ube44\ub85c\uc18c \uc815\ub9ac 1\uc758 \uc138 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\uac00 \uc720\uc77c\ud568\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p class=\"marginbottom0\"><span class=\"theorem\">\ud589\ub82c\uc2dd\uc758 \uc720\uc77c\uc131<\/span><\/p>\n<p>\uc815\ub9ac 1\uc758 \uc138 \uc870\uac74 (i), (ii), (iii)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(\\det : M_n (K) \\rightarrow K\\)\ub294 \uc720\uc77c\ud558\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>(4)\uc640 \uac19\uc774 \uc815\uc758\ub41c \ud589\ub82c\uc2dd\uc740 \uc815\ub9ac 1\uc758 \uc138 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub354\uc6b1\uc774 \uc815\ub9ac 4\ub294 \uc815\ub9ac 1\uc758 \uc138 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\uac00 (4)\uc640 \uc77c\uce58\ud568\uc744 \uc124\uba85\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc815\ub9ac 1\uc758 \uc138 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub294 (4)\ub85c\uc11c \uc720\uc77c\ud558\uac8c \uc815\ud574\uc9c4\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc815\ub9ac 4\uc640 \ub530\ub984\uc815\ub9ac 5\uc758 \uacb0\uacfc\ub85c\uc11c \ud589\ub82c\uc2dd\uc758 \uc720\uc6a9\ud55c \uc131\uc9c8\ub4e4\uc744 \uc774\ub04c\uc5b4\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p class=\"marginbottom0\"><span class=\"theorem\">\uc815\ub9ac 6. (\ud589\ub82c\uc2dd\uc758 \ud589\uc804\uac1c\uc640 \uc5f4\uc804\uac1c)<\/span><\/p>\n<p>\\(A\\)\uac00 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc77c \ub54c, \\(A\\)\uc758 \ud589\ub82c\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ol class=\"parenthesis-roman\">\n<li>\uace0\uc815\ub41c \ud589\ubc88\ud638 \\(i,\\) \\(1\\le i\\le n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\det(A) = \\sum_{j=1}^n (-1)^{i+j} a_{ij} \\det(\\partial_{ij} A).\\]\n<\/li>\n<li>\uace0\uc815\ub41c \uc5f4\ubc88\ud638 \\(j,\\) \\(1\\le j\\le n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\det(A) = \\sum_{i=1}^n (-1)^{i+j} a_{ij} \\det(\\partial_{ij} A).\\]\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\n\ud589\ub82c\uc2dd\uc758 \uae30\ubcf8\uc815\ub9ac(\uc815\ub9ac 1)\uc758 \uc99d\uba85\uc740 \uc784\uc758\uc758 \uc5f4\uc5d0 \uc801\uc6a9 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c \ud589\ub82c\uc2dd\uc774 \uc720\uc77c\ud558\uac8c \uc815\ud574\uc9c0\ubbc0\ub85c (i)\uc758 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub2e4\uc74c\uc73c\ub85c \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uacfc \uc804\uce58\ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc774 \uc77c\uce58\ud558\uba70, \ud589\ub82c\uc758 \ud55c \ud589\uc744 \uae30\uc900\uc73c\ub85c \uc804\uac1c\ud55c \ud589\ub82c\uc2dd\uc740 \uadf8 \uc804\uce58\ud589\ub82c\uc758 \ud55c \uc5f4\uc744 \uae30\uc900\uc73c\ub85c \uc804\uac1c\ud55c \ud589\ub82c\uc2dd\uacfc \uc77c\uce58\ud558\ubbc0\ub85c (ii)\uc758 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"corollary margintop2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 7.<\/span><br \/>\n\uc0bc\uac01\ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc740 \ub300\uac01\uc131\ubd84\uc758 \uacf1\uacfc \uac19\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\n\\(A\\)\uac00 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uace0 \uc544\ub798\uc0bc\uac01\ud589\ub82c\uc774\ub77c\uace0 \ud558\uc790. \\(A\\)\uc758 \uccab\uc9f8 \ud589\uc744 \uae30\uc900\uc73c\ub85c \ud589\ub82c\uc2dd\uc744 \uc804\uac1c\ud558\uba74 \uadf8 \uac12\uc740 \\(A\\)\uc758 \ub300\uac01\uc131\ubd84\uc758 \uacf1\uacfc \uac19\ub2e4. \\(A\\)\uac00 \uc704\uc0bc\uac01\ud589\ub82c\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(A\\)\uc758 \uccab\uc9f8 \uc5f4\uc744 \uae30\uc900\uc73c\ub85c \ud589\ub82c\uc2dd\uc744 \uc804\uac1c\ud558\uba74 \uadf8 \uac12\uc740 \\(A\\)\uc758 \ub300\uac01\uc131\ubd84\uc758 \uacf1\uacfc \uac19\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"corollary margintop2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 8.<\/span><br \/>\n\\(A\\)\uac00 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uace0 \ub2e4\uc74c\uacfc \uac19\uc740 \ud615\ud0dc\ub97c \uac00\uc9c0\uace0 \uc788\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[A= \\left( \\begin{array}{c|c}<br \/>\nP&#038;Q \\\\<br \/>\n\\hline<br \/>\n0&#038;R<br \/>\n\\end{array} \\right).\\tag{8}\\]<br \/>\n\uc5ec\uae30\uc11c \\(P\\)\ub294 \\(m\\times m\\) \ud589\ub82c\uc774\uace0 \\(Q\\)\ub294 \\(m\\times (n-m)\\) \ud589\ub82c\uc774\uba70 \\(R\\)\ub294 \\((n-m)\\times(n-m)\\) \ud589\ub82c\uc774\uace0 \\(0\\)\uc740 \\((n-m)\\times m\\) \uc601\ud589\ub82c\uc774\ub2e4. \uadf8\ub7ec\uba74<br \/>\n\\[\\det(A) = \\det(P) \\det(R)\\tag{9}\\]<br \/>\n\uc774\ub2e4. \ud2b9\ud788 \ud589\ub82c \\(Q\\)\uc758 \uc131\ubd84\uc740 \\(A\\)\uc758 \ud589\ub82c\uc2dd\uc744 \uacc4\uc0b0\ud560 \ub54c \uc544\ubb34\ub7f0 \uc5ed\ud560\uc744 \ud558\uc9c0 \uc54a\ub294\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\\(m\\)\uc5d0 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc801\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc790. \\(m=1\\)\uc778 \uacbd\uc6b0 \\(A\\)\uc758 \uccab\uc9f8 \uc5f4\uc744 \uae30\uc900\uc73c\ub85c \ud589\ub82c\uc2dd\uc744 \uc804\uac1c\ud558\uba74<br \/>\n\\[\\det(A) = a_{11} \\det(\\partial_{11} A) = \\det(P)\\det(R)\\]<br \/>\n\uc774\ubbc0\ub85c (10)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \\(m > 1\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(P\\)\uc758 \ud589\uc758 \uac1c\uc218\uac00 \\(m\\) \ubbf8\ub9cc\uc778 \ubaa8\ub4e0 \uacbd\uc6b0 (10)\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(i > m\\)\uc77c \ub54c \\(a_{i1} = 0\\)\uc774\ubbc0\ub85c, \\(A\\)\uc758 \uccab\uc9f8 \uc5f4\uc744 \uae30\uc900\uc73c\ub85c \ud589\ub82c\uc2dd\uc744 \uc804\uac1c\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\det(A)<br \/>\n&#038;= \\sum_{i=1}^n (-1)^{i+1} a_{i1} \\det(\\partial_{i1} A) \\\\[5pt]<br \/>\n&#038;= \\sum_{i=1}^m (-1)^{i+1} a_{i1} \\det(\\partial_{i1} A) \\\\[5pt]<br \/>\n&#038;= \\sum_{i=1}^m (-1)^{i+1} a_{i1} \\det(\\partial_{i1} P) \\det(R) \\\\[5pt]<br \/>\n&#038;= \\det(P) \\det(R). \\tag*{\\(\\blacksquare\\)}<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\ud589\ub82c\uc758 \uac00\uc5ed\uc131\uacfc \ud589\ub82c\uc2dd\uc758 \uad00\uacc4<\/h2>\n<p>\uc815\uc0ac\uac01\ud589\ub82c \\(A\\)\uc758 \uc5ed\ud589\ub82c\uc774 \uc874\uc7ac\ud560 \ub54c \u2018\\(A\\)\ub294 \uac00\uc5ed\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud558\uace0, \\(A\\)\ub97c \uac00\uc5ed\ud589\ub82c\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud589\ub82c\uc2dd\uc740 \ud589\ub82c\uc758 \uac00\uc5ed\uc131\uacfc \uae4a\uc740 \uad00\uacc4\uac00 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\" style=\"padding-bottom: 0;\">\n<p><span class=\"theorem\">\uc815\ub9ac 9.<\/span><br \/>\n\\(A\\)\uc640 \\(B\\)\uac00 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc77c \ub54c<br \/>\n\\[\\det(AB) = \\det(A) \\det(B) .\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uc774 \uc815\ub9ac\uc758 \uc99d\uba85\uc740 \uc815\ub9ac 4\uc758 \uc99d\uba85\uacfc \ube44\uc2b7\ud558\ub2e4. \\(B\\)\uc758 \uc5f4\ubca1\ud130\ub97c \uc774\uc6a9\ud558\uc5ec \uacf1 \\(AB\\)\ub97c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[AB = (A\\cdot B^1 ,\\, \\cdots ,\\, A\\cdot B^j ,\\, \\cdots ,\\, A \\cdot B^n ).\\]<br \/>\n\ub610\ud55c \\(AB\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[A\\cdot B^j = b_{1j} A^1 + \\cdots + b_{nj} A^n .\\]<br \/>\n\uc815\uc758\uc5ed\uacfc \uacf5\uc5ed\uc774 \ubaa8\ub450 \\(\\left\\{ 1,\\, 2,\\,\\cdots,\\,n\\right\\}\\)\uc778 \ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc744 \\(S\\)\ub77c \ud558\uc790. \uc704 \ub4f1\uc2dd\uacfc \ud589\ub82c\uc2dd\uc758 \uc120\ud615\uc131\uc744 \uc774\uc6a9\ud558\uc5ec \\(AB\\)\uc758 \ud589\ub82c\uc2dd\uc744 \uc804\uac1c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\det(AB) = \\sum_{\\varphi \\in S} b_{\\varphi(1)1} \\cdots b_{\\varphi(n)n} \\det(A^{\\varphi(1)} ,\\,\\cdots,\\, A^{\\varphi(n)}).\\]<br \/>\n\uc5ec\uae30\uc11c \uc2e4\uc81c\ub85c \uc704 \ud569\uc5d0 \uc601\ud5a5\uc744 \uc8fc\ub294 \uac83\uc740 \\(\\varphi\\)\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc77c \ub54c\ubfd0\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f8 \uc218 \uc788\ub2e4.<br \/>\n\\[\\det(AB) = \\sum_{\\pi\\in S_n} b_{\\pi(1)1} \\cdots b_{\\pi(n)n} \\det(A^{\\pi(1)} ,\\,\\cdots,\\, A^{\\pi(n)}).\\]<br \/>\n\uc6b0\ubcc0\uc758 \uac01 \ud56d\uc740 \\(A\\)\uc758 \ud589\ub82c\uc2dd\uc5d0 \\(\\pi\\)\uc758 \ubd80\ud638\ub97c \uacf1\ud55c \uc778\uc218\ub97c \uac00\uc9c4\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \uc2dd\uc758 \uc6b0\ubcc0\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ubcc0\ud615\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\det(AB)<br \/>\n&#038;= \\sum_{\\pi\\in S_n} b_{\\pi(1)1} \\cdots b_{\\pi(n)n} \\sigma(\\pi) \\det(A) \\\\[5pt]<br \/>\n&#038;= \\det(A) \\sum_{\\pi \\in S_n} \\sigma(\\pi) b_{\\pi(1)1} \\cdots b_{\\pi(n)n} \\\\[5pt]<br \/>\n&#038;= \\det(A) \\det(B).\\tag*{\\(\\blacksquare\\)}<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 10.<\/span><br \/>\n\\(A\\in M_n (K)\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc740 \ubaa8\ub450 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis-roman marginbottom0\">\n<li>\uc784\uc758\uc758 \\(\\mathbf{y}\\in K^n\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd \\(A\\mathbf{x} = \\mathbf{y}\\)\ub294 \uc801\uc5b4\ub3c4 \ud558\ub098\uc758 \ud574\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc744 \uc0dd\uc131\ud55c\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \ud589\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc744 \uc0dd\uc131\ud55c\ub2e4.<\/li>\n<li>\ub3d9\ucc28\ubc29\uc815\uc2dd \\(A\\mathbf{x} = \\mathbf{0}\\)\uc740 \uc790\uba85\ud55c \ud574 \\(\\mathbf{x} = \\mathbf{0}\\)\ub9cc\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \ud589\ubca1\ud130\ub4e4\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(\\mathbf{y}\\in K^n\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd \\(A\\mathbf{x} = \\mathbf{y}\\)\ub294 \ub531 \ud558\ub098\uc758 \ud574\ub97c \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc758 \uae30\uc800\ub97c \uc774\ub8ec\ub2e4.<\/li>\n<li>\\(A\\)\uc758 \ud589\ubca1\ud130\ub4e4\uc740 \\(K^n\\)\uc758 \uae30\uc800\ub97c \uc774\ub8ec\ub2e4.<\/li>\n<li>\\(A\\)\ub294 \uac00\uc5ed\uc774\ub2e4. \uc989 \\(A\\in\\operatorname{GL}_n (K)\\)\uc774\ub2e4.<\/li>\n<li>\\(\\det(A) \\ne 0.\\)<\/li>\n<li>\\(\\det(\\,^t\\! A) \\ne 0.\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>(i)\ubd80\ud130 (x)\uae4c\uc9c0\uac00 \ubaa8\ub450 \uc11c\ub85c \ub3d9\uce58\uc784\uc740 \uc55e\uc5d0\uc11c \uc99d\uba85\ud558\uc600\ub2e4. \ub610\ud55c \ub530\ub984\uc815\ub9ac 5\uc5d0 \uc758\ud558\uc5ec \\(\\det(A) = \\det(\\,^t\\! A)\\)\uc774\ubbc0\ub85c (xi)\uc640 (xii)\ub294 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(A\\)\uc758 \uc5f4\ubca1\ud130\ub4e4\uc774 \uc11c\ub85c \uc77c\ucc28\uc885\uc18d\uc774\uba74 \ub530\ub984\uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec \\(\\det(A) = 0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (xi)\u21d2(v)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c (x)\u21d2(xi)\uc744 \uc99d\uba85\ud558\uc790. \\(A\\)\uac00 \uac00\uc5ed\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(A\\)\uc758 \uc5ed\ud589\ub82c \\(B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uc989 \\(AB = I_n\\)\uc774\ub2e4. \uc774\ub54c \uc815\ub9ac 9\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\det(A)\\det(B) = \\det(AB) = \\det(I_n) = 1\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\det(A)\\)\uc758 \uac12\uc740 \\(0\\)\uc774 \ub420 \uc218 \uc5c6\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uccb4 \\(K\\)\uc5d0\uc11c \ub367\uc148\uc5d0 \ub300\ud55c \ud56d\ub4f1\uc6d0\uc744 \uc81c\uc678\ud55c \uc9d1\ud569\uc744 \\(K^*\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc704 \uc815\ub9ac\uc5d0\uc11c (x)\uacfc (xi)\uc774 \ub3d9\uce58\ub77c\ub294 \uc0ac\uc2e4\ub85c\ubd80\ud130 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"corollary margintop2\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 11.<\/span><br \/>\n\\(\\det : \\operatorname{GL}_n (K) \\,\\rightarrow\\, K^*\\)\ub294 \uacf1 \uc5f0\uc0b0\uc5d0 \ub300\ud55c \uad70\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\\(\\det : \\operatorname{GL}_n (K) \\,\\rightarrow\\, K^*\\)\uc758 \ud575\uc740 \ud589\ub82c\uc2dd\uc774 \\(1\\)\uc778 \\(n\\times n\\) \ud589\ub82c\ub4e4\uc758 \uacf5\uac04\uc774\ub2e4. \uc774\ub7ec\ud55c \uacf5\uac04\uc744 \\(n\\)\ucc28 <span class=\"defined\">\ud2b9\uc218\uc120\ud615\uad70<\/span>(special linear group)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\operatorname{SL}_n (K)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"margintop2\">\uc5ec\uc778\uc790\uc640 Cramer \uacf5\uc2dd<\/h2>\n<p>\ub2e4\uc74c\uacfc \uac19\uc740 \ud589\ub82c\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[A = \\left(\\begin{array}{ccc} a_{11} &#038; a_{12} &#038; a_{13} \\\\ a_{21} &#038; a_{22} &#038; a_{23} \\\\ a_{31} &#038; a_{32} &#038; a_{33} \\end{array}\\right).\\]<br \/>\n\uc774\uc81c \ub2e4\uc74c \uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[a_{11} \\det(\\partial_{31}A) &#8211; a_{12} \\det(\\partial_{32}A) + a_{13} \\det(\\partial_{33}A).\\tag{10}\\]<br \/>\n\uc774 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ud589\ub82c \\(A &#8216; \\)\uc758 \uc14b\uc9f8 \ud589\uc744 \uae30\uc900\uc73c\ub85c \ud589\ub82c\uc2dd\uc744 \uc804\uac1c\ud55c \uac83\uacfc \uac19\ub2e4.<br \/>\n\\[A &#8216; = \\left(\\begin{array}{ccc} a_{11} &#038; a_{12} &#038; a_{13} \\\\ a_{21} &#038; a_{22} &#038; a_{23} \\\\ a_{11} &#038; a_{12} &#038; a_{13} \\end{array}\\right).\\]<br \/>\n\\(A &#8216; \\)\uc758 \ub450 \ud589\uc774 \uac19\uc73c\ubbc0\ub85c \\(\\det(A &#8216; )=0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[a_{11} \\det(\\partial_{31}A) &#8211; a_{12} \\det(\\partial_{32}A) + a_{13} \\det(\\partial_{33}A) = \\det(A &#8216; ) = 0\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\((-1)^{i+j}\\det(\\partial_{ij}A)\\)\ub97c \\(a_{ij}\\)\uc758 <span class=\"defined\">\uc5ec\uc778\uc790<\/span>(cofactor)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub85c\uc368 \uc704 \ub4f1\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box\">\n<p>\n\uc815\uc0ac\uac01\ud589\ub82c\uc758 \ud55c \ud589\uc758 \uc131\ubd84\ub4e4\uc744 \ub2e4\ub978 \uace0\uc815\ub41c \ud589\uc758 \uac19\uc740 \uc5f4\uc758 \uc5ec\uc778\uc790\uc640 \uacf1\ud558\uc5ec \ub354\ud55c \uacb0\uacfc\ub294 \\(0\\)\uc774\ub2e4. \ub610\ud55c \ud55c \uc5f4\uc758 \uc131\ubd84\ub4e4\uc744 \ub2e4\ub978 \uace0\uc815\ub41c \uc5f4\uc758 \uac19\uc740 \ud589\uc758 \uc5ec\uc778\uc790\uc640 \uacf1\ud558\uc5ec \ub354\ud55c \uacb0\uacfc\ub294 \\(0\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\\(A\\)\uac00 \\(n\\times n\\) \ud589\ub82c\uc774\uace0 \\(C_{ij}\\)\uac00 \\(a_{ij}\\)\uc758 \uc5ec\uc778\uc790\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ud589\ub82c<br \/>\n\\[\\left(\\begin{array}{cccc}<br \/>\nC_{11} &#038; C_{12} &#038; \\cdots &#038; C_{1n} \\\\<br \/>\nC_{21} &#038; C_{22} &#038; \\cdots &#038; C_{2n} \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\nC_{n1} &#038; C_{n2} &#038; \\cdots &#038; C_{nn}<br \/>\n\\end{array}\\right)\\]<br \/>\n\uc744 \\(A\\)\uc758 <span class=\"defined\">\uc5ec\uc778\uc790\ud589\ub82c<\/span>(matrix of cofactor)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \\(A\\)\uc758 \uc5ec\uc778\uc790\ud589\ub82c\uc758 \uc804\uce58\ud589\ub82c\uc744 \\(A\\)\uc758 <span class=\"defined\">\uc218\ubc18\ud589\ub82c<\/span>(adjoint matrix)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\adj(A)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><br \/>\n\ub2e4\uc74c \ud589\ub82c\uc758 \uc218\ubc18\ud589\ub82c\uc744 \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[A = \\left(\\begin{array}{crr}<br \/>\n3 &#038; 2 &#038; -1 \\\\ 1 &#038; 6 &#038; 3 \\\\ 2 &#038; -4 &#038; 0<br \/>\n\\end{array}\\right).\\]<br \/>\n\uc5ec\uc778\uc790\ub97c \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{array}{lll}<br \/>\nC_{11} =12 , &#038; C_{12} = 6 ,  &#038; C_{13} = -16, \\\\<br \/>\nC_{21} = 4,  &#038; C_{22} = 2 ,  &#038; C_{23} = 16, \\\\<br \/>\nC_{31} = 12, &#038; C_{32} = -10, &#038; C_{33} = 16 .<br \/>\n\\end{array}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \uc5ec\uc778\uc790\ud589\ub82c \\(C\\)\uc640 \uc218\ubc18\ud589\ub82c \\(\\adj(A)\\)\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nC &#038;= \\left(\\begin{array}{rrr}<br \/>\n12 &#038; 6 &#038; -16 \\\\ 4 &#038; 2 &#038; 16 \\\\ 12 &#038; -10 &#038; 16<br \/>\n\\end{array}\\right), \\\\[5pt]<br \/>\n\\adj(A) = \\,^t\\! C &#038;=<br \/>\n\\left(\\begin{array}{rrr}<br \/>\n12 &#038; 4 &#038; 12 \\\\ 6 &#038; 2 &#038; -10 \\\\ -16 &#038; 16 &#038; 16<br \/>\n\\end{array}\\right).<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<div class=\"theorem margintop2\" style=\"padding-bottom: 0;\">\n<p><span class=\"theorem\">\uc815\ub9ac 12.<\/span><br \/>\n\\(A\\)\uac00 \uac00\uc5ed\uc778 \\(n\\)\ucc28 \uc815\uc0ac\uac01\ud589\ub82c\uc774\uba74<br \/>\n\\[A^{-1} = \\frac{1}{\\det(A)} \\adj(A).\\tag{11}\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uba3c\uc800 \\(A\\cdot\\adj (A) = \\det (A) I_n\\)\uc784\uc744 \ubcf4\uc774\uc790. \\(\\adj(A)\\)\uc758 \\(ij\\)-\uc131\ubd84\uc744 \\(C_{ji}\\)\ub85c \ub098\ud0c0\ub0b4\uc790.<br \/>\n\uadf8\ub7ec\uba74 \\(A\\cdot\\adj(A)\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[a_{i1} C_{j1} + a_{i2} C_{j2} + \\cdots + a_{in} C_{jn}.\\tag{12}\\]<br \/>\n\\(i=j\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uc5ec\uc778\uc790\uc640 \uadf8 \uc55e\uc5d0 \uacf1\ud574\uc9c4 \uc131\ubd84\uc774 \\(A\\)\uc758 \uac19\uc740 \ud589\uc73c\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c0\ubbc0\ub85c (3)\uc740 \uadf8 \ud589\uc744 \ub530\ub77c \\(\\det(A)\\)\uc758 \uc5ec\uc778\uc790 \uc804\uac1c\ud55c \uac83\uc774\ub2e4. \\(i\\ne j\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uc5ec\uc778\uc790\uc640 \uadf8 \uc55e\uc5d0 \uacf1\ud574\uc9c4 \uc131\ubd84\uc774 \\(A\\)\uc758 \ub2e4\ub978 \ud589\uc73c\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c0\ubbc0\ub85c \uc55e\uc758 \ub17c\uc758\uc5d0 \uc758\ud558\uc5ec (3)\uc758 \uac12\uc740 \\(0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[A\\cdot\\adj(A) = \\left(\\begin{array}{cccc}<br \/>\n\\det(A) &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; \\det(A) &#038; \\cdots &#038; 0 \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\n0 &#038; 0 &#038; \\cdots &#038; \\det(A)<br \/>\n\\end{array}\\right)<br \/>\n=\\det(A) I_n .\\]<br \/>\n\\(A\\)\uac00 \uac00\uc5ed\ud589\ub82c\uc774\ubbc0\ub85c \\(\\det(A) \\ne 0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \uc2dd\uc758 \uc591\ubcc0\uc744 \\(\\det(A)\\)\ub85c \ub098\ub204\uba74<br \/>\n\\[A \\left( \\frac{1}{\\det(A)} \\adj(A) \\right) = I_n\\]<br \/>\n\uc774\ub2e4. \uc774 \uc2dd\uc73c\ub85c\ubd80\ud130 (11)\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><br \/>\n\uc55e\uc758 \ubcf4\uae30 3\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ud589\ub82c \\(A\\)\uc758 \uc5ed\ud589\ub82c\uc744 \uad6c\ud574 \ubcf4\uc790. \\(\\det(A) = 64\\)\uc774\ubbc0\ub85c<br \/>\n\\[A^{-1} = \\frac{1}{\\det(A)}\\adj(A) =<br \/>\n\\frac{1}{64}<br \/>\n\\left(\\begin{array}{rrr}<br \/>\n12 &#038; 4 &#038; 12 \\\\ 6 &#038; 2 &#038; -10 \\\\ -16 &#038; 16 &#038; 16<br \/>\n\\end{array}\\right).\\]\n<\/p>\n<\/div>\n<p>\uc5ec\uc778\uc790\ub97c \uc774\uc6a9\ud558\uba74 \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd\uc758 \ud574 \uacf5\uc2dd\uc744 \uc27d\uac8c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\" style=\"padding-bottom: 0;\">\n<p class=\"marginbottom0\"><span class=\"theorem\">\uc815\ub9ac 13. (Cramer \uacf5\uc2dd)<\/span><\/p>\n<p>\\(A\\mathbf{x} = \\mathbf{b}\\)\uac00 \\(n\\)\uac1c\uc758 \ubbf8\uc9c0\uc218\ub97c \uac00\uc9c4 \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd\uc774\uace0 \\(\\det(A) \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(A\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc744 \\(\\mathbf{b}\\)\ub85c \ubc14\uafbc \ud589\ub82c\uc744 \\(A_j\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \\(A\\mathbf{x} = \\mathbf{b}\\)\uc758 \ud574\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[x_1 = \\frac{\\det(A_1)}{\\det(A)} ,\\,\\, x_2 = \\frac{\\det(A_2)}{\\det(A)} ,\\,\\, \\cdots ,\\,\\, x_n = \\frac{\\det(A_n)}{\\det(A)} .\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uc815\ub9ac 12\uc758 \uc2dd (11)\uc744 \uc774\uc6a9\ud558\uba74 \\(A\\mathbf{x} = \\mathbf{b}\\)\uc758 \ud574\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\mathbf{x} &#038;= A^{-1} \\mathbf{b} = \\frac{1}{\\det(A)}\\adj(A) \\mathbf{b} \\\\[5pt]<br \/>\n&#038;= \\frac{1}{\\det(A)}<br \/>\n\\left(\\begin{array}{cccc}<br \/>\nC_{11} &#038; C_{21} &#038; \\cdots &#038; C_{n1} \\\\<br \/>\nC_{12} &#038; C_{22} &#038; \\cdots &#038; C_{n2} \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\nC_{1n} &#038; C_{2n} &#038; \\cdots &#038; C_{nn}<br \/>\n\\end{array}\\right)<br \/>\n\\left(\\begin{array}{c}<br \/>\nb_1 \\\\ b_2 \\\\ \\vdots \\\\ b_n<br \/>\n\\end{array}\\right).<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(\\mathbf{x}\\)\uc758 \\(j\\)\uc9f8 \ud589\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[x_j = \\frac{b_1 C_{1j} + b_2 C_{2j} + \\cdots + b_n C_{nj}}{\\det(A)}.\\tag{13}\\]<br \/>\n\uc5ec\uae30\uc11c \\(b_1,\\) \\(b_2 ,\\) \\(\\cdots ,\\) \\(b_n\\)\uc740 \\(\\mathbf{b}\\)\uc758 \uc131\ubd84\uc774\ub2e4. \uc774 \uc2dd\uc758 \uc5ec\uc778\uc790\ub4e4\uc740 \\(A\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc5d0\uc11c \uc628 \uac83\uc774\ubbc0\ub85c, \\(A\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc744 \\(\\mathbf{b}\\)\ub85c \ubc14\uafb8\uc5b4\ub3c4 \uc704 \uc2dd\uc758 \uc5ec\uc778\uc790\ub4e4\uc740 \ubcc0\ud558\uc9c0 \uc54a\ub294\ub2e4. \\(A\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc744 \\(\\mathbf{b}\\)\ub85c \ubc14\uafbc \uacb0\uacfc\ub294 \ud589\ub82c \\(A_j\\)\uac00 \ub418\uba70, (13)\uc758 \ubd84\uc790\ub294 \\(A_j\\)\uc758 \\(j\\)\uc9f8 \uc5f4\uc744 \ub530\ub77c \uc804\uac1c\ud55c \uc5ec\uc778\uc790 \uc804\uac1c\uc640 \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[x_j = \\frac{\\det(A_j )}{\\det(A)}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.<\/span><br \/>\n\ub2e4\uc74c \uc5f0\ub9bd\ubc29\uc815\uc2dd\uc744 \ud480\uc5b4 \ubcf4\uc790.<br \/>\n\\[\\begin{cases} 2x-6y = 1 \\\\ 3x-4y = 5 \\end{cases}\\]<br \/>\nCramer \uacf5\uc2dd\uc744 \ub530\ub77c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nx &#038; = \\frac{\\det\\left( \\begin{array}{cr} 1 &#038; -6 \\\\ 5 &#038; -4 \\end{array} \\right)}{\\det\\left( \\begin{array}{cr} 2 &#038; -6 \\\\ 3 &#038; -4 \\end{array} \\right)}<br \/>\n= \\frac{26}{10} = \\frac{13}{5} , \\\\[5pt]<br \/>\ny &#038; = \\frac{\\det\\left( \\begin{array}{cr} 2 &#038; 1 \\\\ 3 &#038; 5 \\end{array} \\right)}{\\det\\left( \\begin{array}{cr} 2 &#038; -6 \\\\ 3 &#038; -4 \\end{array} \\right)}<br \/>\n= \\frac{7}{10} .<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p>\ub2e4 \uc4f0\uace0 \ub098\uc11c \ubcf4\ub2c8 [\ubcf4\uae30 2]\uac00 \uc5c6\ub124. \uc5b8\uc820\uac00 \ucd94\uac00\ud560 \ubcf4\uae30\ub97c \uc704\ud574 \ubc88\ud638\ub97c \ud558\ub098 \ube44\uc6cc\ub454 \uac83\uc774\ub77c\uace0 \ud574\ub450\uc790.<\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n\n<h2 class=\"margintop2\"><\/h2>\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uacc4\uc218\uc640 \uc0c1\uc218\uac00 \uc2e4\uc218\uc778 \uc774\ucc28\ubc29\uc815\uc2dd\uc774 \uc2e4\uc218 \ubc94\uc704\uc5d0\uc11c \uba87 \uac1c\uc758 \ud574\ub97c \uac16\ub294\uc9c0 \uc54c\uc544\ubcf4\uae30 \uc704\ud574\uc11c\ub294 \ud310\ubcc4\uc2dd\uc758 \ubd80\ud638\ub97c \uc0b4\ud3b4\ubcf4\uba74 \ub41c\ub2e4. \uc774\uc640 \ube44\uc2b7\ud558\uac8c \uc815\uc0ac\uac01\ud589\ub82c\uc758 \uc5ed\ud589\ub82c\uc774 \uc874\uc7ac\ud558\ub294\uc9c0 \uc54c\uc544\ubcf4\ub294 \uacf5\uc2dd\uc774 \uc788\ub294\ub370, \uadf8\uac83\uc774 \ud589\ub82c\uc2dd\uc774\ub2e4. \ud589\ub82c\uc2dd\uc740 \ud2b9\uc815\ud55c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc120\ud615\ubc94\ud568\uc218\ub85c \uc815\uc758\ub420 \uc218\ub3c4 \uc788\ub294\ub370, \uadf8\ub7ec\ud55c \ud568\uc218\ub294 \ud06c\uae30\uac00 \uc791\uc740 \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \ud06c\uae30\uac00 \ud070 \ud589\ub82c\uc758 \ud589\ub82c\uc2dd\uc744 \uacc4\uc0b0\ud558\ub294 \uadc0\ub0a9\uc801\uc778 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \ub610\ud55c \ud589\ub82c\uc2dd\uc740 \ud589\ub82c\uc758 \uac01 \uc131\ubd84\ub4e4\uc744 \uc774\uc6a9\ud558\uc5ec \uc9c1\uc811 \uacc4\uc0b0\ud558\ub294 \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ub420 \uc218\ub3c4 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud589\ub82c\uc2dd\uc758&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[57],"tags":[462,426,464,463,459,457,461,460,458,455,456],"class_list":["post-5643","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","tag-cramer-","tag-426","tag-464","tag-463","tag-459","tag-457","tag-461","tag-460","tag-458","tag-455","tag-456"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5643","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=5643"}],"version-history":[{"count":58,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5643\/revisions"}],"predecessor-version":[{"id":6528,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/5643\/revisions\/6528"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=5643"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=5643"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=5643"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}