{"id":363,"date":"2012-04-07T17:08:25","date_gmt":"2012-04-07T08:08:25","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=363"},"modified":"2021-03-10T20:12:49","modified_gmt":"2021-03-10T11:12:49","slug":"tkr201205","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201205\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2012\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u884c\u5217 \\(A =\\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u304c\u6b21\u306e\u6761\u4ef6 (D) \u3092\u6e80\u305f\u3059\u3068\u3059\u308b.<\/p>\r\n

    \r\n
  1. (D)\u3000\\(A\\) \u306e\u6210\u5206 \\(a , b , c , d\\) \u306f\u6574\u6570\u3067\u3042\u308b. \u307e\u305f, \u5e73\u9762\u4e0a\u306e \\(4\\) \u70b9 \\((0, 0)\\) , \\((a, b)\\) , \\((a+c, b+d)\\) , \\((c, d)\\) \u306f\u9762\u7a4d \\(1\\) \u306e\u5e73\u884c\u56db\u8fba\u5f62\u306e \\(4\\) \u3064\u306e\u9802\u70b9\u3092\u306a\u3059.<\/li>\r\n<\/ol>\r\n

    \\(B = \\left( \\begin{array}{cc} 1 & 1 \\\\ 0 & 1 \\end{array} \\right)\\) \u3068\u304a\u304f. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\u884c\u5217 \\(BA\\) \u3068 \\(B^{-1}A\\) \u3082\u6761\u4ef6 (D) \u3092\u6e80\u305f\u3059\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\\(c=0\\) \u306a\u3089\u3070, \\(A\\) \u306b \\(B\\) , \\(B^{-1}\\) \u306e\u3069\u3061\u3089\u304b\u3092\u5de6\u304b\u3089\u6b21\u3005\u306b\u304b\u3051\u308b\u3053\u3068\u306b\u3088\u308a, \\(4\\) \u500b\u306e\u884c\u5217 \\(\\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right)\\) , \\(\\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & 1 \\end{array} \\right)\\) , \\(\\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right)\\) , \\(\\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & -1 \\end{array} \\right)\\) \u306e\u3069\u308c\u304b\u306b\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

    3. (3)<\/strong>\u3000\\(|a| \\geqq |c| >0\\) \u3068\u3059\u308b. \\(BA\\) , \\(B^{-1}A\\) \u306e\u5c11\u306a\u304f\u3068\u3082\u3069\u3061\u3089\u304b\u4e00\u65b9\u306f, \u305d\u308c\u3092 \\(\\left( \\begin{array}{cc} x & y \\\\ z & w \\end{array} \\right)\\) \u3068\u3059\u308b\u3068\r\n\\[\r\n|x|+|z| \\lt |a|+|c|\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

      \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

      \u5e73\u884c\u56db\u8fba\u5f62\u306e\u9762\u7a4d\u306f \\(|ad-bc|\\) \u3068\u8868\u305b\u308b\u306e\u3067, \u6761\u4ef6 (D) \u306f\u6b21\u306e\u6761\u4ef6 (D') \u3068\u540c\u3058\u3067\u3042\u308b.<\/p>\r\n

        \r\n
      1. (D')\u3000\\(|ad-bc| =1\\) \uff08 \\(a\\) , \\(b\\) , \\(c\\) , \\(d\\) \u306f\u6574\u6570\uff09<\/li>\r\n<\/ol>\r\n

        \\[\r\nBA = \\left( \\begin{array}{cc} 1 & 1 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) =\\left( \\begin{array}{cc} a+c & b+d \\\\ c & d \\end{array} \\right)\r\n\\]\r\n\u6761\u4ef6\u3088\u308a, \\(a+c\\) , \\(b+d\\) \u3082\u6574\u6570\u3067\u3042\u308a\r\n\\[\r\n|(a+c)d -(b+d)c| =|ad-bc| =1\r\n\\]\r\n\u3088\u3063\u3066, \\(BA\\) \u3082\u6761\u4ef6 (D) \u3092\u6e80\u305f\u3057\u3066\u3044\u308b.\r\n\u307e\u305f\r\n\\[\r\nB^{-1}A =\\left( \\begin{array}{cc} 1 & -1 \\\\ 0 & 1 \\end{array} \\right) \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) =\\left( \\begin{array}{cc} a-c & b-d \\\\ c & d \\end{array} \\right)\r\n\\]\r\n\u6761\u4ef6\u3088\u308a, \\(a-c\\) , \\(b-d\\) \u3082\u6574\u6570\u3067\u3042\u308a\r\n\\[\r\n|(a-c)d -(b-d)c| =|ad-bc| =1\r\n\\]\r\n\u3088\u3063\u3066, \\(B^{-1}A\\) \u3082\u6761\u4ef6 (D) \u3092\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p>\r\n

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

        (1)<\/strong> \u306e\u7d50\u679c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nB^n A & = B^{n-1} \\left( \\begin{array}{cc} a+c & b+d \\\\ c & d \\end{array} \\right) \\\\\r\n& = \\cdots = \\left( \\begin{array}{cc} a+nc & b+nd \\\\ c & d \\end{array} \\right) \\quad ... [1] \\\\\r\nB^{-n} A & = B^{-(n-1)} \\left( \\begin{array}{cc} a-c & b-d \\\\ c & d \\end{array} \\right) \\\\\r\n& = \\cdots = \\left( \\begin{array}{cc} a-nc & b-nd \\\\ c & d \\end{array} \\right) \\quad ... [2]\r\n\\end{align}\\]\r\n\\(c=0\\) \u306a\u3089\u3070, \\(|ad| =1\\) \u306a\u306e\u3067\r\n\\[\r\n(a,d) = ( \\pm 1 , \\pm 1 )\r\n\\]\r\n

        \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5834\u5408\u5206\u3051\u3057\u3066, [1] [2] \u3092\u7528\u3044\u308b.<\/p>\r\n

          \r\n

        1. 1*<\/strong>\u3000\\((a,d) =( \\pm 1,1)\\) \u306e\u3068\u304d
          \r\n\\(A =\\left( \\begin{array}{cc} \\pm 1 & b \\\\ 0 & 1 \\end{array} \\right)\\) \u306a\u306e\u3067\r\n\\[\r\nB^{-b} A =\\left( \\begin{array}{cc} \\pm 1 & b-b \\cdot 1 \\\\ 0 & 1 \\end{array} \\right) =\\left( \\begin{array}{cc} \\pm 1 & 0 \\\\ 0 & 1 \\end{array} \\right)\r\n\\]<\/li>\r\n

        2. 2*<\/strong>\u3000\\((a,d) =( \\pm 1,-1)\\) \u306e\u3068\u304d
          \r\n\\(A =\\left( \\begin{array}{cc} \\pm 1 & b \\\\ 0 & -1 \\end{array} \\right)\\) \u306a\u306e\u3067\r\n\\[\r\nB^b A =\\left( \\begin{array}{cc} \\pm 1 & b+b \\cdot (-1) \\\\ 0 & -1 \\end{array} \\right) =\\left( \\begin{array}{cc} \\pm 1 & 0 \\\\ 0 & -1 \\end{array} \\right)\r\n\\]<\/li>\r\n<\/ol>\r\n

          \u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

          \\(BA\\) \u306b\u3064\u3044\u3066, \\(x=a+c\\) , \\(z=c\\) .
          \r\n\\(B^{-1}A\\) \u306b\u3064\u3044\u3066, \\(x=a-c\\) , \\(z=c\\) .
          \r\n\u306a\u306e\u3067\r\n\\[\r\n|a+c|+|c| \\lt |a|+|c| \\quad \\text{\u307e\u305f\u306f} \\quad |a-c|+|c| \\lt |a|+|c|\r\n\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n|a+c| \\lt |a| \\quad \\text{\u307e\u305f\u306f} \\quad |a-c| \\lt |a|\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044.
          \r\n\\(a , c\\) \u306e\u6b63\u8ca0\u306b\u3088\u3063\u3066\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

            \r\n

          1. 1*<\/strong>\u3000\\(a \\geqq c \\gt 0\\) \u306e\u3068\u304d\r\n\\[\r\n|a-c| =a-c \\lt a =|a|\r\n\\]<\/li>\r\n

          2. 2*<\/strong>\u3000\\(a \\gt 0 \\gt c \\geqq -a\\) \u306e\u3068\u304d
            \r\n\\(a+c \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\r\n|a+c| =a+c \\lt a =|a|\r\n\\]<\/li>\r\n

          3. 3*<\/strong>\u3000\\(-a \\geqq c \\gt 0 \\gt a\\) \u306e\u3068\u304d
            \r\n\\(a+c \\leqq 0\\) \u306a\u306e\u3067\r\n\\[\r\n|a+c| =-(a+c) \\lt -a =|a|\r\n\\]<\/li>\r\n

          4. 4*<\/strong>\u3000\\(0 \\gt c \\geqq a\\) \u306e\u3068\u304d\r\n\\[\r\n|a-c| =-(a-c) \\lt -a =|a|\r\n\\]<\/li>\r\n<\/ol>\r\n

            \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u884c\u5217 \\(A =\\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u304c\u6b21\u306e\u6761\u4ef6 (D) \u3092\u6e80\u305f\u3059\u3068\u3059\u308b. (D)\u3000\\(A\\) \u306e\u6210\u5206 \\(a , b … \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":[58],"tags":[139,68],"class_list":["post-363","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2012","tag-tokyo_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/363","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=363"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/363\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=363"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=363"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=363"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}