线代 线性方程组 克莱姆法则 克莱姆法则内容
\u7528\u514b\u83b1\u59c6\u6cd5\u5219\u89e3\u4e0b\u5217\u65b9\u7a0b\u7ec4\u514b\u83b1\u59c6\u6cd5\u5219\uff0c\u53c8\u8bd1\u514b\u62c9\u9ed8\u6cd5\u5219\uff08Cramer's Rule\uff09\u662f\u7ebf\u6027\u4ee3\u6570\u4e2d\u4e00\u4e2a\u5173\u4e8e\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u5b9a\u7406\u3002\u5b83\u9002\u7528\u4e8e\u53d8\u91cf\u548c\u65b9\u7a0b\u6570\u76ee\u76f8\u7b49\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u662f\u745e\u58eb\u6570\u5b66\u5bb6\u514b\u83b1\u59c6\uff081704-1752\uff09\u4e8e1750\u5e74\uff0c\u5728\u4ed6\u7684\u300a\u7ebf\u6027\u4ee3\u6570\u5206\u6790\u5bfc\u8a00\u300b\u4e2d\u53d1\u8868\u7684\u3002\u5176\u5b9e\u83b1\u5e03\u5c3c\u5179\u30141693\u3015\uff0c\u4ee5\u53ca\u9a6c\u514b\u52b3\u6797\u30141748\u3015\u4ea6\u77e5\u9053\u8fd9\u4e2a\u6cd5\u5219\uff0c\u4f46\u4ed6\u4eec\u7684\u8bb0\u6cd5\u4e0d\u5982\u514b\u83b1\u59c6\u3002
1. \u514b\u83b1\u59c6\u6cd5\u5219\u7684\u91cd\u8981\u7406\u8bba\u4ef7\u503c\uff1a\u7814\u7a76\u4e86\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u4e0e\u65b9\u7a0b\u7ec4\u89e3\u7684\u5b58\u5728\u6027\u4e0e\u552f\u4e00\u6027\u5173\u7cfb;\u4e0e\u5176\u5728\u8ba1\u7b97\u65b9\u9762\u7684\u4f5c\u7528\u76f8\u6bd4\uff0c\u514b\u83b1\u59c6\u6cd5\u5219\u66f4\u5177\u6709\u91cd\u5927\u7684\u7406\u8bba\u4ef7\u503c\u3002
2.\u5e94\u7528\u514b\u83b1\u59c6\u6cd5\u5219\u5224\u65ad\u5177\u6709N\u4e2a\u65b9\u7a0b\u3001N\u4e2a\u672a\u77e5\u6570\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3:
\uff081\uff09\u5f53\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u884c\u5217\u5f0f\u4e0d\u7b49\u4e8e\u96f6\u65f6\uff0c\u5219\u65b9\u7a0b\u7ec4\u6709\u89e3\uff0c\u4e14\u5177\u6709\u552f\u4e00\u7684\u89e3\uff1b
\uff082\uff09\u5982\u679c\u65b9\u7a0b\u7ec4\u65e0\u89e3\u6216\u8005\u6709\u4e24\u4e2a\u4e0d\u540c\u7684\u89e3\uff0c\u90a3\u4e48\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u884c\u5217\u5f0f\u5fc5\u5b9a\u7b49\u4e8e\u96f6
(3)\u514b\u83b1\u59c6\u6cd5\u5219\u4e0d\u4ec5\u4ec5\u9002\u7528\u4e8e\u5b9e\u6570\u57df\uff0c\u5b83\u5728\u4efb\u4f55\u57df\u4e0a\u9762\u90fd\u53ef\u4ee5\u6210\u7acb\u3002
3.\u514b\u83b1\u59c6\u6cd5\u5219\u7684\u5c40\u9650\u6027\uff1a
\uff081\uff09\u5f53\u65b9\u7a0b\u7ec4\u7684\u65b9\u7a0b\u4e2a\u6570\u4e0e\u672a\u77e5\u6570\u7684\u4e2a\u6570\u4e0d\u4e00\u81f4\u65f6\uff0c\u6216\u8005\u5f53\u65b9\u7a0b\u7ec4\u7cfb\u6570\u7684\u884c\u5217\u5f0f\u7b49\u4e8e\u96f6\u65f6\uff0c\u514b\u83b1\u59c6\u6cd5\u5219\u5931
\u6548\u3002
\uff082\uff09\u8fd0\u7b97\u91cf\u8f83\u5927\uff0c\u6c42\u89e3\u4e00\u4e2aN\u9636\u7ebf\u6027\u65b9\u7a0b\u7ec4\u8981\u8ba1\u7b97N+1\u4e2aN\u9636\u884c\u5217\u5f0f\u3002
\u514b\u83b1\u59c6\u6cd5\u5219\uff08Cramer's Rule\uff09\u662f\u7ebf\u6027\u4ee3\u6570\u4e2d\u4e00\u4e2a\u5173\u4e8e\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u5b9a\u7406\u3002\u5b83\u9002\u7528\u4e8e\u53d8\u91cf\u548c\u65b9\u7a0b\u6570\u76ee\u76f8\u7b49\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u662f\u745e\u58eb\u6570\u5b66\u5bb6\u514b\u83b1\u59c6\uff081704-1752\uff09\u4e8e1750\u5e74\uff0c\u5728\u4ed6\u7684\u300a\u7ebf\u6027\u4ee3\u6570\u5206\u6790\u5bfc\u8a00\u300b\u4e2d\u53d1\u8868\u7684\u3002
\u57fa\u672c\u4ecb\u7ecd
\u3000\u3000\u5047\u82e5\u6709n\u4e2a\u672a\u77e5\u6570\uff0cn\u4e2a\u65b9\u7a0b\u7ec4\u6210\u7684\u65b9\u7a0b\u7ec4\uff1a \u514b\u83b1\u59c6\u6cd5\u5219(9\u5f20)
\u3000\u3000a11X1+a12X2+\uff0e\uff0e\uff0e+a1nXn = b1,
\u3000\u3000a21X1+a22X2+\uff0e\uff0e\uff0e+a2nXn = b2,
\u3000\u3000\uff0e\uff0e\uff0e\uff0e\uff0e\uff0e
\u3000\u3000an1X1+an2X2+\uff0e\uff0e\uff0e+annXn = bn.
\u3000\u3000\u6216\u8005\u5199\u6210\u77e9\u9635\u5f62\u5f0f\u4e3aAx=b\uff0c\u5176\u4e2dA\u4e3an*n\u65b9\u9635\uff0cx\u4e3an\u4e2a\u53d8\u91cf\u6784\u6210\u5217\u5411\u91cf\uff0cb\u4e3an\u4e2a\u5e38\u6570\u9879\u6784\u6210\u5217\u5411\u91cf\u3002
\u3000\u3000\u800c\u5f53\u5b83\u7684\u7cfb\u6570\u77e9\u9635\u53ef\u9006\uff0c\u6216\u8005\u8bf4\u5bf9\u5e94\u7684\u884c\u5217\u5f0f|A|\u4e0d\u7b49\u4e8e0\u7684\u65f6\u5019\uff0c\u5b83\u6709\u552f\u4e00\u89e3xi=|Ai|/|A|\uff0c\u5176\u4e2dAi\u3014i = 1\uff0c2\uff0c\u2026\u2026\uff0cn\u3015\u662f\u77e9\u9635A\u4e2d\u7b2ci\u5217\u7684a 1i\uff0ca 2i\uff0c\u2026\u2026a ni \uff08\u5373\u7b2ci\u5217\uff09\u4f9d\u6b21\u6362\u6210b1\uff0cb2\uff0c\u2026\u2026bn\u6240\u5f97\u7684\u77e9\u9635\u3002
\u3000\u3000\u514b\u83b1\u59c6\u6cd5\u5219\u4e0d\u4ec5\u4ec5\u9002\u7528\u4e8e\u5b9e\u6570\u57df\uff0c\u5b83\u5728\u4efb\u4f55\u57df\u4e0a\u9762\u90fd\u53ef\u4ee5\u6210\u7acb\u3002
\u3000\u3000\u4f7f\u7528\u514b\u83b1\u59c6\u6cd5\u5219\u6c42\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u7684\u7b97\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230O(n^3\uff09\uff0c\u8fd9\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\u540c\u5176\u5b83\u5e38\u7528\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u65b9\u6cd5\uff0c\u6bd4\u5982\u9ad8\u65af\u6d88\u5143\u6cd5\u76f8\u5f53\u3002
\u3000\u3000\u5f53b1,b2,...,bn\u4e0d\u5168\u4e3a0\u65f6\uff0c\u65b9\u7a0b\u7ec4\u4e3a\u975e\u9f50\u6b21\u6027\u65b9\u7a0b\u7ec4\u3002
\u3000\u3000\u7cfb\u6570\u77e9\u9635A\u975e\u5947\u5f02\u65f6\uff0c\u6216\u8005\u8bf4\u884c\u5217\u5f0f|A|\u22600\u65f6\uff0c\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u7684\u89e3\uff1b
\u3000\u3000\u7cfb\u6570\u77e9\u9635A\u5947\u5f02\u65f6\uff0c\u6216\u8005\u8bf4\u884c\u5217\u5f0f|A|=0\u65f6\uff0c\u65b9\u7a0b\u7ec4\u6709\u65e0\u6570\u4e2a\u89e3\u3002
\u3000\u3000\u5f53b1=b2=...=bn=0\u65f6\uff0c\u65b9\u7a0b\u7ec4\u4e3a\u9f50\u6b21\u6027\u65b9\u7a0b\u7ec4\u3002
\u3000\u3000\u82e5\u7cfb\u6570\u77e9\u9635A\u975e\u5947\u5f02\u65f6\uff0c\u5219\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u7684\u89e3\uff0c\u5176\u6240\u6709\u5206\u91cf\u5747\u4e3a0\uff0c\u6211\u4eec\u901a\u5e38\u79f0\u8fd9\u4e2a\u89e3\u4e3a\u5e73\u51e1\u89e3\u3002
\u3000\u3000\u82e5\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u975e\u96f6\u89e3\uff0c\u7cfb\u6570\u77e9\u9635\u5fc5\u7136\u5947\u5f02\uff0c\u6216\u8005\u8bf4\u5bf9\u5e94\u7684\u7cfb\u6570\u884c\u5217\u5f0f\u5fc5\u4e3a0\u3002
\u3000\u3000\u5176\u5b9e\u83b1\u5e03\u5c3c\u5179\u30141693\u3015\uff0c\u4ee5\u53ca\u9a6c\u514b\u52b3\u6797\u30141748\u3015\u4ea6\u77e5\u9053\u8fd9\u4e2a\u6cd5\u5219\uff0c\u4f46\u4ed6\u4eec\u7684\u8bb0\u6cd5\u4e0d\u5982\u514b\u83b1\u59c6\u3002
\u6cd5\u5219\u603b\u7ed3
\u3000\u30001\uff1a\u514b\u83b1\u59c6\u6cd5\u5219\u7684\u91cd\u8981\u7406\u8bba\u4ef7\u503c\uff1a\u7814\u7a76\u4e86\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u4e0e\u65b9\u7a0b\u7ec4\u89e3\u7684\u5b58\u5728\u6027\u4e0e\u552f\u4e00\u6027\u5173\u7cfb;
\u3000\u30002\uff1a\u5e94\u7528\u514b\u83b1\u59c6\u6cd5\u5219\u5224\u65ad\u5177\u6709N\u4e2a\u65b9\u7a0b\u3001N\u4e2a\u672a\u77e5\u6570\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3:
\u3000\u3000\uff081\uff09\uff1a\u5f53\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u884c\u5217\u5f0f\u4e0d\u7b49\u4e8e\u96f6\u65f6\uff0c\u5219\u65b9\u7a0b\u7ec4\u6709\u89e3\uff0c\u4e14\u5177\u6709\u552f\u4e00\u7684\u89e3\uff1b
\u3000\u3000\uff082\uff09\uff1a\u5982\u679c\u65b9\u7a0b\u7ec4\u65e0\u89e3\u6216\u8005\u6709\u4e24\u4e2a\u4e0d\u540c\u7684\u89e3\uff0c\u90a3\u4e48\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u884c\u5217\u5f0f\u5fc5\u5b9a\u7b49\u4e8e\u96f6\uff1b
\u3000\u30003\uff1a\u514b\u83b1\u59c6\u6cd5\u5219\u7684\u5c40\u9650\u6027\uff1a
\u3000\u3000\uff081\uff09\uff1a\u5f53\u65b9\u7a0b\u7ec4\u7684\u65b9\u7a0b\u4e2a\u6570\u4e0e\u672a\u77e5\u6570\u7684\u4e2a\u6570\u4e0d\u4e00\u81f4\u65f6\uff0c\u6216\u8005\u5f53\u65b9\u7a0b\u7ec4\u7cfb\u6570\u7684\u884c\u5217\u5f0f\u7b49\u4e8e\u96f6\u65f6\uff0c\u514b\u83b1\u59c6\u6cd5\u5219\u5931
\u3000\u3000\u6548\u3002
\u3000\u3000\uff082\uff09\uff1a\u8fd0\u7b97\u91cf\u8f83\u5927\uff0c\u6c42\u89e3\u4e00\u4e2aN\u9636\u7ebf\u6027\u65b9\u7a0b\u7ec4\u8981\u8ba1\u7b97N+1\u4e2aN\u9636\u884c\u5217\u5f0f\u3002
\u6280\u672f\u5e94\u7528
\u3000\u3000\u514b\u83b1\u59c6\u6cd5\u5219\u5728\u89e3\u51b3\u5fae\u5206\u51e0\u4f55\u65b9\u9762\u5341\u5206\u6709\u7528\u3002
\u3000\u3000\u5148\u8003\u8651\u4e24\u6761\u7b49\u5f0f\u548c\u3002\u56e0\u4e3au\u548cv\u90fd\u662f\u6ca1\u76f8\u5173\u7684\u53d8\u6570\uff0c\u6211\u4eec\u53ef\u5b9a\u4e49\u548c\u3002
\u3000\u3000\u627e\u51fa\u4e00\u6761\u7b49\u5f0f\u9002\u5408\u662f\u514b\u83b1\u59c6\u6cd5\u5219\u7684\u7b80\u5355\u5e94\u7528\u3002
\u3000\u3000\u9996\u5148\uff0c\u6211\u4eec\u8981\u8ba1\u7b97F\u3001G\u3001x\u548cy\u7684\u5bfc\u6570\uff1a
\u3000\u3000\u5c06dx\u548cdy\u4ee3\u5165dF\u548cdG\uff0c\u53ef\u5f97\u51fa\uff1a
\u3000\u3000\u56e0\u4e3au\u548cv\u90fd\u6ca1\u6709\u5173\u7cfb\uff0c\u6240\u4ee5du\u548cdv\u7684\u7cfb\u6570\u90fd\u8981\u7b49\u4e8e0\u3002\u6240\u4ee5\u7b49\u5f0f\u4e2d\u7684\u7cfb\u6570\u53ef\u4ee5\u88ab\u5199\u6210\uff1a
\u3000\u3000\u73b0\u5728\u7528\u514b\u83b1\u59c6\u6cd5\u5219\u5c31\u53ef\u5f97\u5230\uff1a
\u3000\u3000\u7528\u4e24\u4e2a\u96c5\u53ef\u6bd4\u77e9\u9635\u6765\u8868\u793a\u7684\u65b9\u7a0b\uff1a
\u3000\u3000\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u5c31\u53ef\u4ee5\u627e\u5230\u3001\u4ee5\u53ca\u3002
\u4ee5\u4e0a\u5185\u5bb9\u6765\u81ea\u767e\u5ea6\u767e\u79d1\u3002\u3002\u3002\u3002\u3002\u3002
D = |A^T| = |A| = (a2-a1)(a3-a1)(a4-a1)(a3-a2)(a4-a2)(a4-a3)
根据克莱姆法则， D1 = D2 = D3 = 0， D4 = D
则 x1 = x2 = x3 = 0, x4 = 1
扩展阅读：matlab验证克莱姆法则 ... 用克莱姆法则解方程 ... 克莱姆法则运算公式 ... 克拉默法则求解公式 ... 克拉默法则的运算公式 ... 克莱姆法则结论 ... 克莱姆法则公式 图文 ... 克莱姆法则怎么用 ... 克莱姆法则求解方程组公式 ...
