四阶行列式展开有几项 按照行列式的定义展开4阶行列式,共有多少项?为什么

\u56db\u9636\u884c\u5217\u5f0f\u7684\u5b8c\u5168\u5c55\u5f00\u5f0f\u5171\u6709\u591a\u5c11\u9879

\u56db\u9636\u884c\u5217\u5f0f\u7684\u5b8c\u5168\u5c55\u5f00\u5f0f\u5171\u670924\u9879\uff01\u8fc7\u7a0b\u5982\u4e0b\uff1a
1\u3001\u56db\u9636\u884c\u5217\u5f0f\u5c55\u5f00\uff0c\u5171\u67094\u4e2a\u4e0d\u540c\u7684\u4e09\u9636\u884c\u5217\u5f0f\uff1b
2\u3001\u6309\u3010\u884c\u5217\u5f0f\u5c55\u5f00\u5b9a\u7406\u3011,4\u9636\u884c\u5217\u5f0f\u5c55\u5f00\u6210\u4f4e\u4e00\u9636\u7684\u4e09\u9636\u884c\u5217\u5f0f\u65f6,\u6709\u56db\u4e2a\u5206\u884c\u5217\u5f0f\uff1b\u7ee7\u7eed\u3010\u5c55\u5f00\u3011\u4e0b\u53bb,\u6bcf\u4e2a3\u9636\u884c\u5217\u5f0f\u53ef\u4ee5\u3010\u5c55\u3011\u62103\u4e2a2\u9636\u884c\u5217\u5f0f\uff1b\u6bcf\u4e2a2\u9636\u884c\u5217\u5f0f\u53ef\u4ee5\u3010\u5c55\u3011\u62102\u9879.\u6240\u4ee5\u5168\u90e8\u5c55\u5f00\u540e\u5171\u6709 4!=24\u9879\u2014\u2014\u548c\u5b9a\u4e49\u63cf\u8ff0\u7684\u76f8\u540c!D4=a11A11+a12A12+a13A13+a14A14=a11M11-a12M12+a13M13-a14M14

\u62d3\u5c55\u8d44\u6599\uff1a
1\u3001\u6309\u7167\u4e00\u5b9a\u7684\u89c4\u5219\uff0c\u7531\u6392\u6210\u6b63\u65b9\u5f62\u7684\u4e00\u7ec4(n\u4e2a)\u6570(\u79f0\u4e3a\u5143\u7d20)\u4e4b\u4e58\u79ef\u5f62\u6210\u7684\u4ee3\u6570\u548c\uff0c\u79f0\u4e3an\u9636\u884c\u5217\u5f0f\u3002
\u4f8b\u5982\uff0c\u56db\u4e2a\u6570a\u3001b\u3001c\u3001d\u6240\u6392\u6210\u4e8c\u9636\u884c\u5f0f\u8bb0\u4e3a
\uff0c\u5b83\u7684\u5c55\u5f00\u5f0f\u4e3aad-bc\u3002
\u4e5d\u4e2a\u6570a1\uff0ca2\uff0ca3;b1\uff0cb2\uff0cb3;c1\uff0cc2\uff0cc3\u6392\u6210\u7684\u4e09\u9636\u884c\u5217\u5f0f\u8bb0\u4e3a
\uff0c\u5b83\u7684\u5c55\u5f00\u5f0f\u4e3aa1b2c3+a2b3c1+a3b1c2-a1b3c2-a2b1c3-a3b2c1\u3002
2\u3001\u884c\u5217\u5f0f\u8d77\u6e90\u4e8e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u6c42\u89e3\uff0c\u5728\u6570\u5b66\u5404\u5206\u652f\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002\u5728\u4ee3\u6570\u4e0a\uff0c\u884c\u5217\u5f0f\u53ef\u7528\u6765\u7b80\u5316\u67d0\u4e9b\u8868\u8fbe\u5f0f\uff0c\u4f8b\u5982\u8868\u793a\u542b\u8f83\u5c11\u672a\u77e5\u6570\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u7b49\u3002

\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1\uff1an\u9636\u884c\u5217\u5f0f

\u6839\u636e\u5b9a\u4e49\uff1an\u9636\u884c\u5217\u5f0f\u7531n!\u4e2a\uff08n\u4e2a\u5143\u7d20\u4e58\u79ef\u7684\uff09\u9879\u7ec4\u6210\u3002\u6240\u4ee5 4 \u9636\u884c\u5217\u5f0f\u670924\u9879\u3002

四阶行列式完全展开有24项。

六项

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