大学线性代数，行列式求解 大学线性代数，计算这个行列式

\u7ebf\u6027\u4ee3\u65704\u9636\u884c\u5217\u5f0f\u7684\u8ba1\u7b97

r2-r1*2\u3001r3-r1\u3001r4-r1*2
D4=|1 2 1 2|
0 -3 -1 -3
0 -1 0 0
0 -3 0 -3
\u63d0\u51far2\u516c\u56e0\u5b50(-1)\uff1b\u4ea4\u6362c2\u3001c3
=|1 1 2 2|
0 1 3 3
0 0 -1 0
0 0 -3 -3
\u4ea4\u6362r3\u3001r4\uff1b\u4ea4\u6362c3\u3001c4
=|1 1 2 2|
0 1 3 3
0 0 -3 -3
0 0 0 -1 \u3010\u8fd9\u662f\u4e2a\u300a\u4e0a\u4e09\u89d2\u300b\u3011
=1*1*\uff08-3\uff09*\uff08-1\uff09
=3


\u62d3\u5c55\u8d44\u6599\uff1a\u7ebf\u6027\u4ee3\u6570\u662f\u6570\u5b66\u7684\u4e00\u4e2a\u5206\u652f\uff0c\u5b83\u7684\u7814\u7a76\u5bf9\u8c61\u662f\u5411\u91cf\uff0c\u5411\u91cf\u7a7a\u95f4\uff08\u6216\u79f0\u7ebf\u6027\u7a7a\u95f4\uff09\uff0c\u7ebf\u6027\u53d8\u6362\u548c\u6709\u9650\u7ef4\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u7ebf\u6027\u4ee3\u6570\u7684\u7406\u8bba\u662f\u8ba1\u7b97\u6280\u672f\u7684\u57fa\u7840\uff0c\u540c\u7cfb\u7edf\u5de5\u7a0b\uff0c\u4f18\u5316\u7406\u8bba\u53ca\u7a33\u5b9a\u6027\u7406\u8bba\u7b49\u6709\u7740\u5bc6\u5207\u8054\u7cfb\uff0c\u968f\u7740\u8ba1\u7b97\u6280\u672f\u7684\u53d1\u5c55\u548c\u8ba1\u7b97\u673a\u7684\u666e\u53ca\uff0c\u7ebf\u6027\u4ee3\u6570\u4f5c\u4e3a\u7406\u5de5\u79d1\u7684\u4e00\u95e8\u57fa\u7840\u8bfe\u7a0b\u65e5\u76ca\u53d7\u5230\u91cd\u89c6\u3002\u7ebf\u6027\u4ee3\u6570\u8fd9\u95e8\u8bfe\u7a0b\u7684\u7279\u70b9\u662f\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6982\u5ff5\u4e4b\u95f4\u8054\u7cfb\u5f88\u5bc6\u5207\u3002\u5185\u5bb9\u5305\u62ec\u884c\u5217\u5f0f\uff0c\u77e9\u9635\uff0c\u5411\u91cf\u7a7a\u95f4\uff0c\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u77e9\u9635\u7684\u76f8\u4f3c\u5bf9\u89d2\u5316\uff0c\u4e8c\u6b21\u578b\uff0c\u7ebf\u6027\u7a7a\u95f4\u4e0e\u7ebf\u6027\u53d8\u6362\u7b49\u3002\u5c5e\u4e8e\u5927\u5b66\u4e00\u5e74\u7ea7\u5de5\u79d1\u90e8\u5206\u8ba1\u7b97\u673a\u53ca\u7535\u6c14\uff0c\u7ecf\u7ba1\u7c7b\u4e13\u4e1a\u5b66\u751f\u5fc5\u4fee\u79d1\u76ee\uff0c\u4e5f\u53ef\u4f9b\u79d1\u6280\u5de5\u4f5c\u8005\u9605\u8bfb\u3002\u7ebf\u6027\u4ee3\u6570\u7684\u7406\u8bba\u5df2\u88ab\u6cdb\u5316\u4e3a\u7b97\u5b50\u7406\u8bba\u3002\u7531\u4e8e\u79d1\u5b66\u7814\u7a76\u4e2d\u7684\u975e\u7ebf\u6027\u6a21\u578b\u901a\u5e38\u53ef\u4ee5\u88ab\u8fd1\u4f3c\u4e3a\u7ebf\u6027\u6a21\u578b\uff0c\u4f7f\u5f97\u7ebf\u6027\u4ee3\u6570\u88ab\u5e7f\u6cdb\u5730\u5e94\u7528\u4e8e\u81ea\u7136\u79d1\u5b66\u548c\u793e\u4f1a\u79d1\u5b66\u4e2d\u3002

\u659c\u5411\u76f8\u4e58\u51cf\u53bb\u53cd\u659c\u5411\u76f8\u4e58\uff1a2\u00d77\u00d72\u00d72+\uff08-5\uff09\u00d7\uff08-1\uff09\u00d7\uff087\uff09+1\u00d74+2+\uff08-3\uff09\u00d7\uff08-9\uff09\u00d71+5\u00d7\uff08-6\uff09+4-4\u00d7\uff08-9\uff09\u00d7\uff08-1\uff09\u00d72-\uff08-6\uff09\u00d72\u00d74+1\u00d77+2-5\u00d77\u00d71-\uff08-3\uff09\u00d7\uff08-5\uff09-2=13
\u5316\u7b80\u53cd\u800c\u589e\u52a0\u96be\u5ea6\uff0c\u6709\u5206\u6570\u3002

将第n行全部替换为1，即为所求的代数余子式之和：
然后前n-1行，都减去最后1行的a倍，
可以化成下三角行列式，即等于
(x-a)^(n-1)

用定义搞定
第2,3,4行取法唯一
只有两项非零
D= a^2 + (-1)^t(52341)e^2 = a^2 - e^2

绾挎т唬鏁,姹傝鍒楀紡
绛旓細a^2 a 1/a 1 1/a^2 a 1/a 1 b^2 b 1/b 1 1/b^2 b 1/b 1 c^2 c 1/c 1 + 1/c^2 c 1/

澶у 绾挎т唬鏁 琛屽垪寮 璺眰澶х
绛旓細鎸夌涓鍒楀睍寮锛屽緱鍒颁袱涓猲-1闃琛屽垪寮锛堝叾涓1涓槸涓嬩笁瑙掞紝鍙︿竴涓浣淒n-1锛夊嵆绛変簬a1x1x2x3...xn-1+y1Dn-1 灏咲n-1缁х画鎸夌1鍒楀睍寮锛岀被浼煎緱鍒颁袱涓猲-2闃惰鍒楀紡锛堝叾涓1涓槸涓嬩笁瑙掞紝鍙︿竴涓浣淒n-2锛夊嵆绛変簬a1x1x2x3...xn-1+y1(a2x2x3...xn-1+y2Dn-2)濡傛杩涜涓嬪幓锛屽緱鍒 a1x1x2x...

澶у鏁板绾挎т唬鏁 琛屽垪寮闂,鏈涢珮鎵嬭В绛?
绛旓細1銆丄₂₁绛変簬(-1)¹⁺²涔樹互濡備笅琛屽垪寮锛殀 -5 2 1 | | 3 1 3 | | -4 -1 -3 |锛岀浜岃鍔犲埌绗笁琛岋紝寰楋細| -5 2 1 | | 3 1 3 | | -1 0 0 |锛屾寜绗笁琛屽睍寮锛屽緱锛-1*(6-1)=-5 鎵浠₂₁=(-1)*(-5)=5 A₃&#...

绾挎т唬鏁棰樼洰,璁＄畻涓嬪垪琛屽垪寮
绛旓細绗紙1锛夐锛屾寜鐓х1鍒楀睍寮锛屽緱鍒癮*琛屽垪寮1 +锛-1锛塣(n+1)*琛屽垪寮2 鍏朵腑琛屽垪寮1鏄瑙掗樀琛屽垪寮忥紝绛変簬a^(n-1)琛屽垪寮2锛堟槸n-1闃讹級锛屽啀鎸夌1琛屽睍寮锛屽緱鍒帮紙-1锛塣n*琛屽垪寮3 鍏朵腑琛屽垪寮3鏄瑙掗樀锛岀瓑浜巃^(n-2)鍥犳锛屾渶缁堢粨鏋滄槸 a*a^(n-1) + 锛-1锛塣(n+1)*锛-1锛塣n*a^(n-2)...

绾挎т唬鏁,鐨琛屽垪寮鎬庝箞姹傚憿
绛旓細浣犺繖涓煩闃礎鍙互鐪嬫垚鏄4涓2脳2鐭╅樀鎷艰捣鏉ョ殑锛岀敱浜庡彸涓婅鐨2脳2鐭╅樀涓0鐭╅樀锛屾晠鐢卞垎鍧楃煩闃电殑琛屽垪寮璁＄畻鍙緱det(A)=(1脳4-2脳3)脳(5脳8-6脳7)=(-2)脳(-2)=4

澶у绾挎т唬鏁拌鍒楀紡闂
绛旓細绗1锛1锛夐锛屽厛鎻愬彇绗2鍒楀叕鍥犲瓙锛屽皢鍒嗘暟鍙樻垚鏁存暟銆傜2锛2锛夐锛屾媶寮绗1鍒楋紝鍙樻垚涓や釜琛屽垪寮忎箣鍜岋紙鍏朵腑涓涓琛屽垪寮忥紝绗1鍒楅兘涓1锛夋帴涓嬫潵锛屼娇鐢ㄥ垵绛夊彉鎹紝鍖栦负涓嬩笁瑙掞紝鍙︿竴涓鍒楀紡锛屾寜绗1鍒楀睍寮锛屽嵆鍙緱鍒颁綆1闃剁殑琛屽垪寮忥紝鐒跺悗濡傛硶鐐埗锛屽嵆鍙 绗3锛1锛夐锛屾媶寮绗1鍒楋紝鎶宸х被浼间笂涓棰 ...

绾挎т唬鏁,璁＄畻琛屽垪寮
绛旓細瑙ｏ細鍘熷紡= 1 2 -1 0 -1 4 5 -1 2 3 1 3 3 1 -2 0 渚濇灏嗙浜岃鍔犱笂绗竴琛岋紝绗笁琛屽噺鍘荤涓琛岀殑 2 鍊嶏紝绗洓琛屽噺鍘荤涓琛岀殑 3 鍊嶏紝寰 1 2 -1 0 0 6 4 -1 0 -1 3 3 0 -5 1 0 绗笁琛...

绾挎т唬鏁,杩欎釜琛屽垪寮鏄庝箞姹傜殑鍛,鎯宠璇︾粏杩囩▼?
绛旓細-1 1 2 1 1 2 3 2 2 3 -> 1 1 -1 1 0-1 3 0 (绗簩琛屽噺绗竴琛岀殑涓ゅ嶏級0-1` 5 0 锛堢涓夎鍑忕涓琛岀殑3鍊嶏級-> 1 1 -1 1 0-1 3 0 0 0 2 0 (绗笁琛屽噺绗簩琛, 浠庤屽緱鍒板瑙掔嚎宸︿笅鐨勫厓绱犵殕涓洪浂銆傦級-> x3 = 0; x2 = 0, x1 = 1 Answer: (1, 0, 0)

澶у绾挎т唬鏁,琛屽垪寮忔眰瑙
绛旓細灏嗙n琛屽叏閮ㄦ浛鎹负1锛屽嵆涓烘墍姹鐨浠ｆ暟浣欏瓙寮忎箣鍜岋細鐒跺悗鍓峮-1琛岋紝閮藉噺鍘绘渶鍚1琛岀殑a鍊嶏紝鍙互鍖栨垚涓嬩笁瑙琛屽垪寮忥紝鍗崇瓑浜 (x-a)^(n-1)

楂樼瓑鏁板绾挎т唬鏁,璇烽棶涓闃琛屽垪寮鐨勫艰鎬庝箞璁＄畻
绛旓細涓闃琛屽垪寮鐢变竴涓暟缁勬垚锛屽畠鐨勫煎氨鏄繖涓暟鏈韩銆備竴闃惰鍒楀紡灏辨槸浠呮湁涓琛屼竴鍒楃殑琛屽垪寮 涓闃惰鍒楀紡灏辩瓑浜庡畠鐨勫厓绱 鎹㈣█涔嬶紝|a|=a

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