有关线性代数的转置和逆矩阵的问题 线性代数问题，A不是方阵，怎么求它的转置的啊？还有不是方阵的...
\u7ebf\u6027\u4ee3\u6570\u4e2d\u7684\u77e9\u9635\u7684\u8f6c\u7f6e\u548c\u77e9\u9635\u7684\u9006\u77e9\u9635\u6709\u4ec0\u4e48\u533a\u522b\u548c\u8054\u7cfb\uff1f\u4e00\u3001\u7ebf\u6027\u4ee3\u6570\u4e2d\u7684\u77e9\u9635\u7684\u8f6c\u7f6e\u548c\u77e9\u9635\u7684\u9006\u77e9\u9635\u67092\u70b9\u4e0d\u540c\uff1a
1\u3001\u4e24\u8005\u7684\u542b\u4e49\u4e0d\u540c\uff1a
\uff081\uff09\u77e9\u9635\u8f6c\u7f6e\u7684\u542b\u4e49\uff1a\u5c06A\u7684\u6240\u6709\u5143\u7d20\u7ed5\u7740\u4e00\u6761\u4ece\u7b2c1\u884c\u7b2c1\u5217\u5143\u7d20\u51fa\u53d1\u7684\u53f3\u4e0b\u65b945\u5ea6\u7684\u5c04\u7ebf\u4f5c\u955c\u9762\u53cd\u8f6c\uff0c\u5373\u5f97\u5230A\u7684\u8f6c\u7f6e\u3002\u4e00\u4e2a\u77e9\u9635M, \u628a\u5b83\u7684\u7b2c\u4e00\u884c\u53d8\u6210\u7b2c\u4e00\u5217\uff0c\u7b2c\u4e8c\u884c\u53d8\u6210\u7b2c\u4e8c\u5217\u7b49\uff0c\u6700\u672b\u4e00\u884c\u53d8\u4e3a\u6700\u672b\u4e00\u5217\uff0c \u4ece\u800c\u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u77e9\u9635N\u3002 \u8fd9\u4e00\u8fc7\u7a0b\u79f0\u4e3a\u77e9\u9635\u7684\u8f6c\u7f6e\u3002\u5373\u77e9\u9635A\u7684\u884c\u548c\u5217\u5bf9\u5e94\u4e92\u6362\u3002
\uff082\uff09\u9006\u77e9\u9635\u7684\u542b\u4e49\uff1a\u4e00\u4e2an\u9636\u65b9\u9635A\u79f0\u4e3a\u53ef\u9006\u7684\uff0c\u6216\u975e\u5947\u5f02\u7684\uff0c\u5982\u679c\u5b58\u5728\u4e00\u4e2an\u9636\u65b9\u9635B\uff0c\u4f7f\u5f97AB=BA=E\uff0c\u5219\u79f0B\u662fA\u7684\u4e00\u4e2a\u9006\u77e9\u9635\u3002A\u7684\u9006\u77e9\u9635\u8bb0\u4f5cA-1\u3002
2\u3001\u4e24\u8005\u7684\u57fa\u672c\u6027\u8d28\u4e0d\u540c\uff1a
\uff081\uff09\u77e9\u9635\u8f6c\u7f6e\u7684\u57fa\u672c\u6027\u8d28\uff1a(A\u00b1B)T=AT\u00b1BT\uff1b(A\u00d7B)T= BT\u00d7AT\uff1b(AT)T=A\uff1b(KA)T=KA\u3002
\uff082\uff09\u9006\u77e9\u9635\u7684\u57fa\u672c\u6027\u8d28\uff1a\u53ef\u9006\u77e9\u9635\u4e00\u5b9a\u662f\u65b9\u9635\u3002\u5982\u679c\u77e9\u9635A\u662f\u53ef\u9006\u7684\uff0c\u5176\u9006\u77e9\u9635\u662f\u552f\u4e00\u7684\u3002A\u7684\u9006\u77e9\u9635\u7684\u9006\u77e9\u9635\u8fd8\u662fA\u3002\u8bb0\u4f5c\uff08A-1\uff09-1=A\u3002\u53ef\u9006\u77e9\u9635A\u7684\u8f6c\u7f6e\u77e9\u9635AT\u4e5f\u53ef\u9006\uff0c\u5e76\u4e14\uff08AT\uff09-1=\uff08A-1\uff09T (\u8f6c\u7f6e\u7684\u9006\u7b49\u4e8e\u9006\u7684\u8f6c\u7f6e\uff09\u3002
\u4e8c\u3001\u77e9\u9635\u7684\u8f6c\u7f6e\u548c\u9006\u77e9\u9635\u4e4b\u95f4\u7684\u8054\u7cfb\uff1a\u77e9\u9635\u7684\u8f6c\u7f6e\u548c\u9006\u77e9\u9635\u662f\u4e24\u4e2a\u5b8c\u5168\u4e0d\u540c\u7684\u6982\u5ff5\u3002\u8f6c\u7f6e\u662f\u884c\u53d8\u6210\u5217\u5217\u53d8\u6210\u884c\uff0c\u6ca1\u6709\u672c\u8d28\u7684\u53d8\u6362\uff0c\u9006\u77e9\u9635\u662f\u548c\u77e9\u9635\u7684\u8f6c\u7f6e\u76f8\u4e58\u4ee5\u540e\u6210\u4e3a\u5355\u4f4d\u77e9\u9635\u7684\u77e9\u9635\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u4e00\u3001\u9006\u77e9\u9635\u7684\u5176\u5b83\u6027\u8d28\uff1a
1\u3001\u82e5\u77e9\u9635A\u53ef\u9006\uff0c\u5219\u77e9\u9635A\u6ee1\u8db3\u6d88\u53bb\u5f8b\u3002\u5373AB=O\uff08\u6216BA=O\uff09\uff0c\u5219B=O\uff0cAB=AC\uff08\u6216BA=CA\uff09\uff0c\u5219B=C\u3002
2\u3001\u4e24\u4e2a\u53ef\u9006\u77e9\u9635\u7684\u4e58\u79ef\u4f9d\u7136\u53ef\u9006\u3002
3\u3001\u77e9\u9635\u53ef\u9006\u5f53\u4e14\u4ec5\u5f53\u5b83\u662f\u6ee1\u79e9\u77e9\u9635\u3002
\u4e8c\u3001\u9006\u77e9\u9635\u6027\u8d28\u7684\u8bc1\u660e\uff1a
1\u3001\u9006\u77e9\u9635\u662f\u5bf9\u65b9\u9635\u5b9a\u4e49\u7684\uff0c\u56e0\u6b64\u9006\u77e9\u9635\u4e00\u5b9a\u662f\u65b9\u9635\u3002\u8bbeB\u4e0eC\u90fd\u4e3aA\u7684\u9006\u77e9\u9635\uff0c\u5219\u6709B=C\u3002
2\u3001\u5047\u8bbeB\u548cC\u5747\u662fA\u7684\u9006\u77e9\u9635\uff0cB=BI=B\uff08AC\uff09=\uff08BA\uff09C=IC=C\uff0c\u56e0\u6b64\u67d0\u77e9\u9635\u7684\u4efb\u610f\u4e24\u4e2a\u9006\u77e9\u9635\u76f8\u7b49\u3002
3\u3001\u7531\u9006\u77e9\u9635\u7684\u552f\u4e00\u6027\uff0cA-1\u7684\u9006\u77e9\u9635\u53ef\u5199\u4f5c\uff08A-1\uff09-1\u548cA\uff0c\u56e0\u6b64\u76f8\u7b49\u3002
4\u3001\u77e9\u9635A\u53ef\u9006\uff0c\u6709AA-1=I \u3002\uff08A-1\uff09TAT=\uff08AA-1\uff09T=IT=I \uff0cAT\uff08A-1\uff09T=\uff08A-1A\uff09T=IT=I\u7531\u53ef\u9006\u77e9\u9635\u7684\u5b9a\u4e49\u53ef\u77e5\uff0cAT\u53ef\u9006\uff0c\u5176\u9006\u77e9\u9635\u4e3a\uff08A-1\uff09T\u3002\u800c\uff08AT\uff09-1\u4e5f\u662fAT\u7684\u9006\u77e9\u9635\uff0c\u7531\u9006\u77e9\u9635\u7684\u552f\u4e00\u6027\uff0c\u56e0\u6b64\uff08AT\uff09-1=\uff08A-1\uff09T\u3002
5\u3001\u5728AB=O\u4e24\u7aef\u540c\u65f6\u5de6\u4e58A-1\uff08BA=O\u540c\u7406\u53ef\u8bc1\uff09\uff0c\u5f97A-1\uff08AB\uff09=A-1O=O\uff0c\u800cB=IB=\uff08AA-1\uff09B=A-1\uff08AB\uff09\uff0c\u6545B=O\u3002
6\u3001\u7531AB=AC\uff08BA=CA\u540c\u7406\u53ef\u8bc1\uff09\uff0cAB-AC=A(B-C)=O\uff0c\u7b49\u5f0f\u4e24\u8fb9\u540c\u5de6\u4e58A-1\uff0c\u56e0A\u53ef\u9006AA-1=I \u3002\u5f97B-C=O\uff0c\u5373B=C\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u77e9\u9635\u8f6c\u7f6e
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u8f6c\u7f6e
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u77e9\u9635\u8f6c\u7f6e
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u9006\u77e9\u9635
2012\u8003\u7814\u771f\u9898\u3002\u3002\u3002\u6c42\u8f6c\u7f6e\u548c\u662f\u4e0d\u662f\u65b9\u9635\u6ca1\u5173\u7cfb\u7684\uff0c\u6240\u6709\u77e9\u9635\u90fd\u6709\u8f6c\u7f6e\uff0c\u53ea\u6709\u65b9\u9635\u624d\u6709\u53ef\u9006\u77e9\u9635\u3002
这里要用到一个定理：若两个方阵A与B的乘积是单位阵，则A与B互为逆矩阵。利用运算性质改写等式可以得出A的简化表达式。
下图的解答要点请你参考。
绛旓細杩欓噷瑕佺敤鍒颁竴涓畾鐞嗭細鑻ヤ袱涓柟闃礎涓嶣鐨勪箻绉槸鍗曚綅闃碉紝鍒橝涓嶣浜掍负閫嗙煩闃銆傚埄鐢ㄨ繍绠楁ц川鏀瑰啓绛夊紡鍙互寰楀嚭A鐨勭畝鍖栬〃杈惧紡銆備笅鍥剧殑瑙ｇ瓟瑕佺偣璇蜂綘鍙傝冦
绛旓細杞疆锛屽拰姹傞嗙殑鏃跺欙紝鐩镐箻鐨鐭╅樀鏄棰犲掍竴涓嬩綅缃殑銆
绛旓細A(I-C'B)T =CT' ('琛ㄧず閫嗭級涓や晶鍚屾椂杞疆寰楀埌 (I-C'B)AT =C'涓や晶鍚屾椂宸︿箻(I-C'B)鐨勯嗗緱鍒 AT= (I-C'B)'C'鍚屾椂杞疆寰楀埌A=C'T(I-C'B)'T 杩欐牱瑙ｇ瓟瑕佹眰鎵鏈鐭╅樀閮藉彲閫嗭紝涓擨-C'B鍙嗭紝浣犻鐩腑骞舵病鏈夛紝鎵浠ラ鐩槸鏈夌憰鐤电殑 ...
绛旓細涓銆绾挎т唬鏁涓殑鐭╅樀鐨勮浆缃拰鐭╅樀鐨勯嗙煩闃鏈2鐐逛笉鍚岋細1銆佷袱鑰呯殑鍚箟涓嶅悓锛氾紙1锛鐭╅樀杞疆鐨勫惈涔夛細灏咥鐨勬墍鏈夊厓绱犵粫鐫涓鏉′粠绗1琛岀1鍒楀厓绱犲嚭鍙戠殑鍙充笅鏂45搴︾殑灏勭嚎浣滈暅闈㈠弽杞紝鍗冲緱鍒癆鐨勮浆缃備竴涓煩闃礛, 鎶婂畠鐨勭涓琛屽彉鎴愮涓鍒楋紝绗簩琛屽彉鎴愮浜屽垪绛夛紝鏈鏈竴琛屽彉涓烘渶鏈竴鍒楋紝 浠庤屽緱鍒颁竴涓...
绛旓細涓嶈.婊¤冻 A-1=AT 鐨勬槸缁欏畾鐨勬弧瓒充竴瀹氭潯浠剁殑A, 鑰屼笉涓浠绘剰鐨凙閮芥弧瓒 鎶夾=A*浠ｅ叆 鐩稿綋浜嶢*婊¤冻 A•AT=E .浜嬪疄涓婃湁:1. (A*)^T = (A^T)2. (AB)* = B*A 鎵浠 鐢 AA^T = E 鍙緱 (A*)^TA* = E 鎵浠 (A*)^-1 = (A*)^T.涓嶈繃閭ｄ袱涓粨璁鸿瘉璧锋潵涔熼夯鐑 ...
绛旓細IAI=2锛寍A|=|A^t|锛寍<A^(-1)>^t|=|锛圓^t锛塣(-1)|=|A^(-1)|=1/|A|=1/2
绛旓細杩欐槸涓涓垎鍧楀瑙掔煩闃甸棶棰橈紝瀵逛簬鍒嗗潡瀵硅鐭╅樀锛屽彧瑕佺湅瀹冪殑姣忎釜瀵硅瀛愬潡鏄惁鍙嗭紝濡傛灉鍙嗭紝鍒欏彧闇姹傚嚭姣忎釜瀛愬潡鐨閫嗙煩闃灏卞彲浠ヤ簡銆傝棰樹腑锛屽緢瀹规槗鐪嬪嚭锛屾瘡涓瑙掑瓙鍧楃殑琛屽垪寮忛兘涓嶄负0锛屾晠閮藉彲閫嗭紝浜庢槸鍙姹備袱涓簩闃鐭╅樀鐨閫嗗氨鍙互浜嗐傝屼簩闃剁煩闃垫眰閫嗗彲浠ョ敤涓ゆ崲涓闄ょ洿鎺ュ啓鍑烘潵銆傛晠鐭╅樀鐨勯嗕负锛...
绛旓細鐭╅樀閫嗙殑杞疆锛屾槸杞疆鍚庣殑閫嗙煩闃銆傚洜姝ゆ槸(x'x)'鐨勯嗙煩闃碉紝涔熷嵆x'x锛坸'x鏄绉扮煩闃碉級鐨勯嗙煩闃
绛旓細鍥犱负r(A)=n,鎵浠鏄彲閫嗙殑瀹炲绉扮煩闃碉紝浜庢槸A閫嗙煩闃电殑杞疆=A鐨勮浆缃殑閫嗙煩闃=A鐨勯嗙煩闃 鎵浠鐨勯嗙煩闃典篃鏄疄瀵圭О鐭╅樀锛屽嵆A*/涓ˋ涓ㄤ篃鏄绉扮殑锛岄偅涔圓*涔熸槸瀹炲绉扮煩闃 閭ｄ箞Aij=Aji锛屾墍浠ュ氨寰楀埌浣犳瀛愰噷妗嗚捣鏉ョ殑杩欎竴閮ㄥ垎浜嗐傜函鎵嬫墦锛屾湜閲囩撼銆
绛旓細杞疆鏄妸鐭╅樀鐨琛屽彉涓哄垪銆佸垪鍙樹负琛岋紝鏃犺鏄笉鏄柟闃碉紝閮藉彲浠ヨ浆缃閫嗙煩闃鏄笌鍘熺煩闃电殑绉瓑浜庡崟浣嶇煩闃电殑鐭╅樀銆備粎鏂归樀鎵嶅彲鑳藉瓨鍦ㄩ嗙煩闃点
