{"id":796,"date":"2013-05-17T01:17:54","date_gmt":"2013-05-16T16:17:54","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=796"},"modified":"2021-09-24T17:37:28","modified_gmt":"2021-09-24T08:37:28","slug":"tok201302","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201302\/","title":{"rendered":"\u6771\u5de5\u59272013\uff1a\u7b2c2\u554f"},"content":{"rendered":"
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\\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306b\u5bfe\u3057\u3066, \\(\\mathit{\\Delta} (A) = ad-bc\\) , \\(t(A) = a+d\\) \u3068\u5b9a\u3081\u308b.<\/p>\r\n

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  1. (1)<\/strong>\u3000\\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217 \\(A , B\\) \u306b\u5bfe\u3057\u3066, \\(\\mathit{\\Delta} (AB) = \\mathit{\\Delta} (A) \\mathit{\\Delta} (B)\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(A\\) \u306e\u6210\u5206\u304c\u3059\u3079\u3066\u5b9f\u6570\u3067, \\(A^5 = E\\) \u304c\u6210\u308a\u7acb\u3064\u3068\u304d, \\(x = \\mathit{\\Delta} (A)\\) \u3068 \\(y = t(A)\\) \u306e\u5024\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(E\\) \u306f \\(2\\) \u6b21\u306e\u5358\u4f4d\u884c\u5217\u3068\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

    (1)<\/strong><\/p>\r\n

    \\(B = \\left( \\begin{array}{cc} p & q \\\\ r & s \\end{array} \\right)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\mathit{\\Delta} (B) = ps -qr\r\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nAB & = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{cc} p & q \\\\ r & s \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} ap+br & aq+bs \\\\ cp+dr & cq+ds \\end{array} \\right)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\mathit{\\Delta} (AB) & = (ap+br) (cq+ds) -(aq+bs) (cp+dr) \\\\\r\n& = adps +bcrs -adqr -bcps \\\\\r\n& = (ad-bc)(ps-qr) \\\\\r\n& = \\mathit{\\Delta} (A) \\mathit{\\Delta} (B)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\mathit{\\Delta} (AB) = \\mathit{\\Delta} (A) \\mathit{\\Delta} (B)\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070, \\(A^5 = E \\quad ... [1]\\) \u3088\u308a\r\n\\[\r\nx^5 = 1\r\n\\]\r\n\\(A\\) \u306e\u6210\u5206\u304c\u3059\u3079\u3066\u5b9f\u6570\u306a\u306e\u3067, \\(x\\) \u3082\u5b9f\u6570\u3060\u304b\u3089\r\n\\[\r\nx = \\underline{1}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u30b1\u30fc\u30ea\u30fc\u30fb\u30cf\u30df\u30eb\u30c8\u30f3\u306e\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\nA^2-yA+E & = O \\\\\r\n\\text{\u2234} \\quad A^2 & = yA-E\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\nA^5 & = ( yA-E )^2 A \\\\\r\n& = y^2 ( yA-E ) A -2y ( yA-E ) -A \\\\\r\n& = y^3 ( yA-E ) -(3y^2+1) A +2yE \\\\\r\n& = ( y^4-3y^2+1 ) A -(y^3-2y) E\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 [1] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\r\n( y^4-3y^2+1 ) A = ( y^3-2y+1 ) E \\quad ... [2]\r\n\\]\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(A = kE\\) \uff08 \\(k\\) \u306f\u5b9f\u6570\uff09\u306e\u3068\u304d
      \r\n[1] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\r\n(k^5-1) E = OE\r\n\\]\r\n\\(k\\) \u306f\u5b9f\u6570\u306a\u306e\u3067\r\n\\[\r\nk=1\r\n\\]\r\n\u3053\u306e\u3068\u304d\r\n\\[\r\ny = 1+1 = 2\r\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(A \\neq kE\\) \uff08 \\(k\\) \u306f\u5b9f\u6570\uff09\u306e\u3068\u304d
      \r\n[2] \u3088\u308a\r\n\\[\r\n\\left\\{\\begin{array}{ll} y^4-3y^2+1 = 0 & \\quad ... [3] \\\\ y^3-2y+1 = 0 & \\quad ... [4] \\end{array}\\right.\r\n\\]\r\n[4] \u3088\u308a, \\(y^3 = 2y-1\\) \u306a\u306e\u3067, [3] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\ny ( 2y-1 ) -3y^2+1 & = 0 \\\\\r\ny^2+y-1 & = 0 \\\\\r\n\\text{\u2234} \\quad y & = \\dfrac{-1 \\pm \\sqrt{5}}{2}\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b \\(y\\) \u306e\u5024\u306f\r\n\\[\r\ny = \\underline{2 , \\ \\dfrac{-1 \\pm \\sqrt{5}}{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306b\u5bfe\u3057\u3066, \\(\\mathit{\\Delta} (A) = … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[92],"tags":[141,111],"class_list":["post-796","post","type-post","status-publish","format-standard","hentry","category-toko_2013","tag-toko","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/796","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=796"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/796\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=796"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=796"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}