设A为3阶实对称矩阵，且满足A³=A，二次型f(x)=X'AX的正负惯性指数都是1，则|3A+2E 设n阶实对称矩阵A的秩为r，且满足A2=A，求①二次型xTA...
A\u4e3a3\u9636\u5b9e\u5bf9\u79f0\u77e9\u9635\uff0c\u4e14\u6ee1\u8db3A^3-A^2-A=2E,\u4e8c\u6b21\u578bA^TAx\u7ecf\u6b63\u4ea4\u53d8\u6362\u53ef\u5316\u4e3a\u6807\u51c6\u578b\uff0c\u6c42\u6b64\u6807\u8bbe g(t)=t^3-t^2-t-2
\u5219g(t)\u662f\u77e9\u9635A\u7684\u5316\u96f6\u591a\u9879\u5f0f
g(t)=(t-2)(t^2+t+1)
\u56e0\u4e3aA\u662f\u662f\u5bf9\u79f0\u77e9\u9635\uff0c\u7279\u5f81\u503c\u90fd\u4e3a\u5b9e\u6570
\u6240\u4ee5 \u7279\u5f81\u503c t=2
\u4e8e\u662f x^T A^T Ax \u7684\u6807\u51c6\u578b\u4e3a 4x^Tx
\u8bbeA\u03b1=\u03bb\u03b1\uff08\u03b1\u22600\uff09\uff0c\u5219A2\u03b1=\u03bb2\u03b1\uff0c\u53c8A2\u03b1=A\u03b1=\u03bb\u03b1\uff0c\u6545\u03bb2\u03b1=\u03bb\u03b1?\uff08\u03bb2-\u03bb\uff09\u03b1=0?\u03bb=1\u6216\u8005\u03bb=0\uff0e\u7531n\u9636\u5b9e\u5bf9\u79f0\u77e9\u9635A\u7684\u79e9\u4e3ar\u77e5\uff0c\u03bb=1\uff0c\u03bb=0\u5206\u522b\u4e3aA\u7684r\u91cd\u548cn-r\u91cd\u7279\u5f81\u503c\uff0c\u6545\u5b58\u5728\u6b63\u4ea4\u77e9\u9635P\uff0c\u4f7f\u5f97P?1AP\uff1dPTAP\uff1dErOOO\uff0e\u2460\u7ecf\u6b63\u4ea4\u53d8\u6362x=Py\uff0c\u4e8c\u6b21\u578bxTAx\u7684\u6807\u51c6\u5f62\u4e3ay21+y22+\u2026+y2r\uff0e\u2461A2=A?A2=\u2026=An=A\uff0c \u4ee4A=P\u2227P-1\u6545\u884c\u5217\u5f0f|E+nA|=|PP-1+nP\u039bP-1||P\uff08E+n\u039b\uff09P-1|=|P||E+n\u039b||P-1|=|E+n\u039b|=\uff08n+1\uff09r\uff0e
可利用惯性指数确定三个特征值，从而求出行列式为-10。经济数学团队帮你解答，请及时采纳。谢谢！
绛旓細濡備笅鍥炬墍绀
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绛旓細璁疚鏄疉鐨勭壒寰佸 鍒 位^2+2位 鏄 A^2+2A 鐨勭壒寰佸 鑰 A^2=2A = 0 鎵浠 位^2+2位 = 0 鎵浠 位=0 鎴 位 = -2.鍗矨鐨勭壒寰佸煎彧鑳芥槸 0 鎴 -2.鍥犱负 r(2E+A) = 2 鎵浠 A 鐨勫睘浜庣壒寰佸-2鐨勭嚎鎬ф棤鍏崇殑鐗瑰緛鍚戦噺鏈 3-2=1 涓 鎵浠 -2 鏄疉鐨勫崟閲嶆牴 鎵浠 A鐨勭壒寰佸间负 0,...
绛旓細璁続鏄3闃跺疄瀵圭О鐭╅樀,婊¤冻A鈭2=3A,涓擱(A)=2,閭ｄ箞鐭╅樀A鐨勪笁涓壒寰佸兼槸? 鎴戞潵绛 棣栭〉 鐢ㄦ埛 璁よ瘉鐢ㄦ埛 瑙嗛浣滆 甯府鍥 璁よ瘉鍥㈤槦 鍚堜紮浜 浼佷笟 濯掍綋 鏀垮簻 鍏朵粬缁勭粐 鍟嗗煄 娉曞緥 鎵嬫満绛旈 鎴戠殑 璁続鏄3闃跺疄瀵圭О鐭╅樀,婊¤冻A鈭2=3A,涓擱(A)=2,閭ｄ箞鐭╅樀A鐨勪笁涓壒寰佸兼槸? 鎴戞潵绛 ...
绛旓細A²+3A=0 鏁匒(A+3E)=0锛屾晠A鍙湁鐗瑰緛鍊0鍜-3,鏈夊洜涓簉(A)=2 鏁匒鐨勭壒寰佸间负-3锛-3,0 A+kE鐨勭壒寰佸间负k-3锛宬-3,k 鑰孉+kE鐨勬槸姝ｅ畾鐨勫厖瑕佹潯浠舵槸浠栫殑鐗瑰緛鍊煎潎澶т簬闆躲傛晠k>3鏃讹紝A+kE涓烘瀹氱煩闃点傛敞锛氭湰棰樿瘉鏄庝緷璧A鏄疄涓夐樁瀵圭О鐭╅樀銆
绛旓細鍥犱负宸茬煡A2+2A=0锛屾墍浠鐨勭壒寰佸兼槸0鎴-2锛岄偅涔坘A鐨勭壒寰佸兼槸0鎴-2k锛宬A+E鐨勭壒寰佸兼槸1鎴1-2k锛庡張鐢辨瀹氱殑鍏呭垎蹇呰鏉′欢鏄壒寰佸煎叏澶т簬0锛A鏄3闃跺疄瀵圭О闃碉紝鎵浠1?锛1-2k锛夛紴0锛屾墍浠锛12锛屾晠绛旀涓猴細锛12锛
绛旓細鍥犱负A鍙浉浼煎瑙掑寲 鎵浠涓庡瑙鐭╅樀B鐩镐技,涓擝鐨勪富瀵硅绾夸笂鐨勫厓绱犻兘鏄疉鐨勭壒寰佸 鑰岀浉浼肩煩闃电殑绉╃浉鍚 鎵浠ュ瑙掔煩闃礏鐨勭З涔熸槸涓2 鎵浠鐨勯潪闆剁壒寰佸肩殑涓暟涓2 鏁呯壒寰佸间负 0,-2,-2 鎬荤粨:鍙瑙掑寲鐨勭煩闃电殑绉 绛変簬 鐭╅樀闈為浂鐗瑰緛鍊肩殑涓暟 ...
绛旓細璁綼鏄疉鐨勪换涓鐗瑰緛鍚戦噺 鍒欙紙A^3-A^2-A-2E锛塧=(位^3-位^2-位-2)a=(位-2)(位^2+位+1)a=0 鍥犱负a鏄瀹炲绉扮煩闃,鐗瑰緛鍊煎叏涓哄疄鏁 鎵浠ノ=2 鎵浠鐨勭壒寰佸煎叏涓2 鎵浠鏍囧噯褰负2x1^2+2x2^2+2x3^2
绛旓細璁 x涓轰换涓鐗瑰緛鍚戦噺锛宺涓哄搴旂壒寰佹牴銆A^2=A ==> A2x=Ax ==> (r^2-r)x=0 ==> r(r-1)=0 鎵浠 r=1 鎴 0 鍥犱负 R(A)=2, 鎵浠ョ壒寰佹牴蹇呯劧鏄 1锛1锛0
绛旓細A2=A鏄浠涔堬紵鎵撻敊浜嗗惂锛岄夯鐑︿慨鏀逛竴涓嬨傚鏋滄槸A^2=A 鍗矨^2-A=0 鍐欐垚鐗瑰緛鍊兼柟绋嬑籢2-位=0 鎵浠鍙兘鐨勭壒寰佸兼槸锛0鍜1 鍥犱负A鐨勭З鏄2锛屾墍浠ユ槸1,1,0 鏂规硶鎬荤粨涓涓嬪氨鏄 --- 鐢ㄧ粰鐨鐭╅樀鍏崇郴寮忥紝鍐欏嚭鐗瑰緛鍊兼柟绋嬶紝鐒跺悗瑙ｅ嚭鍙兘鐨勭壒寰佸硷紝杩欎簺鐗瑰緛鍊煎彧鏄彲鑳藉硷紝鏈夊嚑涓 锛屾湁娌℃湁閮芥槸涓嶇‘瀹...
