A为n阶方阵,A*是A的伴随矩阵,已知A的各行元素之和为2,A*的各行元素之和为3，则|A|=？，急求~谢谢大家了~ 设n阶方阵A的每行,每列元素之和都是0。证明:A的伴随矩阵A...
\u8bben\u9636\u77e9\u9635A\u7684\u5404\u5217\u5143\u7d20\u4e4b\u548c\u4e3a2\uff0c\u4e14|A|=6\uff0c\u5219\u5b83\u7684\u4f34\u968f\u77e9\u9635A*\u7684\u5404\u5217\u5143\u7d20\u4e4b\u548c\u4e3a\uff08\u3000\u3000\uff09A\uff0e2B\uff0e13C\uff0e3D\uff0e\u7531\u4e8e\u77e9\u9635A\u7684\u5404\u5217\u5143\u7d20\u4e4b\u548c\u4e3a2\uff0c\u6240\u4ee5\uff0cAT11?1=211?1\u7531A*A\uff1d.A.E\u53ef\u77e5\uff1a\uff08A*\uff09TAT=|A|E=6E\u6545\u6709\uff1a\uff08A*\uff09TAT11?1=2\uff08A*\uff09T11?1=611?1\u5f97\uff1a\uff08A*\uff09T11?1=311?1\u6545\u9009\u62e9\uff1aC\uff0e
\u4f60\u597d\uff01\u53ef\u4ee5\u5229\u7528\u77e9\u9635\u7684\u79e9\u4ee5\u53ca\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u57fa\u7840\u89e3\u7cfb\u6765\u8bc1\u660e\uff0c\u8981\u70b9\u5982\u56fe\u3002\u7ecf\u6d4e\u6570\u5b66\u56e2\u961f\u5e2e\u4f60\u89e3\u7b54\uff0c\u8bf7\u53ca\u65f6\u91c7\u7eb3\u3002\u8c22\u8c22\uff01
将|B|按第一列展开，故|A|等于2乘以|B|的第一列的代数余子式的和。
注意到|B|的第一列的代数余子式和|A|的第一列的代数余子式相同，其和恰为，A*的第一行的元素的元素的和为3.
绛旓細鎵浠A*鍙,(A*)^-1 = A/|A| (A^-1)(A^-1)* = E/|A| 涓よ竟鍚屾椂宸︿箻A (A^-1)* = A/|A| = (A*)^-1
绛旓細鑰岀煩闃A鐨绉涓簄-1,鎵浠ヨ鍦ˋ涓殑n-1闃瀛愬紡涓嚦灏戞湁涓涓笉涓0锛屾墍浠A*涓湁鍏冪礌涓嶄负0锛屽嵆A*鈮0锛宺(A*)>=1銆傛墍浠 r(A*)=1 褰搑(A)<n-1,鍗硆(A)<=n-2鏃讹紝璇存槑鐭╅樀鐨勭З鏄皬浜庣瓑浜巒-2鐨勶紝閭ｄ箞浠栫殑鎵浠-1闃跺瓙寮忓叏涓0锛屽氨鏄A*涓殑姣忎釜鍏冪礌鍏ㄤ负0.A*=0.鎵浠(A*)=0 ...
绛旓細灏唡A|鐨勬瘡涓鍒楀姞鍏ョ涓鍒,绗竴鍒楀彉涓2,鎻愬嚭2,寰楀埌2|B|銆 灏唡B|鎸夌涓鍒楀睍寮,鏁厊A|绛変簬2涔樹互|B|鐨勭涓鍒楃殑浠ｆ暟浣欏瓙寮忕殑鍜屻 娉ㄦ剰鍒皘B|鐨勭涓鍒楃殑浠ｆ暟浣欏瓙寮忓拰|A|鐨勭涓鍒楃殑浠ｆ暟浣欏瓙寮忕浉鍚,鍏跺拰鎭颁负,A*鐨绗竴琛岀殑鍏冪礌鐨勫厓绱犵殑鍜屼负3. 宸茶禐杩 宸茶俯杩< 浣犲杩欎釜鍥炵瓟鐨勮瘎浠锋槸? 璇勮 鏀惰捣...
绛旓細鍥犱负锛欰A*=A*A=|A|E锛屾墍浠ュ浜庝换鎰弉缁村悜閲徫辨湁锛AA*伪锛.A.E伪锛5伪鏁咃細|A|=5鏄疊=AA*鐨鐗瑰緛鍊硷紝浠绘剰n缁撮潪闆跺悜閲徫卞彲涓哄搴旂殑鐗瑰緛鍚戦噺锛
绛旓細鈭A*=A^-1*|A| 鈭磡A*|=|A^-1*|A|| =|A|^n *|A^-1| =|A|^n *|A|^-1 -|A|^(n-1)閫塁D C涓嶥绛旀涓鏍凤紵锛燂紵
绛旓細AA*=|A|E 涓よ竟鍙栬鍒楀紡 |AA*|=| |A|E| 寰梶A||A*|=|A|^n|E|=|A|^n 鍥犳|A*|=|A|^(n-1)=a^(n-1)
绛旓細璁A鏄痭闃舵柟闃碉紝A*鏄疉鐨勪即闅忕煩闃碉紝A*=0璇存槑A鐨勪换浣曚唬鏁颁綑瀛愬紡涓0锛屼篃灏辨槸浠讳綍涓涓猲-1闃跺瓙寮忎负0锛屾墍浠(A)<n-1銆
绛旓細|A|=0锛宺(A)锛n銆俽(A*)鈮0锛屾墍浠(A*)=1锛宺(A)=n-1 閫氳В涓簁尾1锛宬鈮0
绛旓細鑻(A)<n-1,鍒橝涓墍鏈塶-1闃瀛愬紡鍧囦负0锛屽嵆琛屽垪寮弢A|鐨勬墍鏈変唬鏁颁綑瀛愬紡鍧囦负0锛屽嵆琛屽垪寮弢A|鐨勬墍鏈変唬鏁颁綑瀛愬紡鍧囦负0锛屽嵆A*=O锛屾晠r(A*)=0.
绛旓細B=AA*=|A|E= 2 ..2 ...2 ...2 n闃 鎵浠ョ壒寰佸间负2(n閲)鐗瑰緛鍚戦噺涓何1=(0,0,0...0,0,1)^T,伪2=(0,0,0...0,1,0)^T,伪3=(0,0,0...1,0,0)^T...伪n=(1,0,0...0,0,0)^T
