证明：奇数阶反对称矩阵的行列式为零． 证明奇数级反对称阵的行列式为0
\u8bc1\u660e\uff1a\u5947\u6570\u9636\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u884c\u5217\u5f0f\u4e3a\u96f6\u8bc1\u660e\uff1a\u6839\u636e\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u6027\u8d28\u6709\uff1a
AT=-A
|A|=|AT|=|-A|=\uff08-1\uff09n|A|=-|A|
\u7531\u4e8en\u4e3a\u5947\u6570
\u6240\u4ee5|A|=0
\u8bbeA\u4e3an\u7ef4\u65b9\u9635\uff0c\u82e5\u6709A'=-A\uff0c\u5219\u79f0\u77e9\u9635A\u4e3a\u53cd\u5bf9\u79f0\u77e9\u9635\u3002\u5bf9\u4e8e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u5b83\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5168\u4e3a\u96f6\uff0c\u800c\u4f4d\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u4e24\u4fa7\u5bf9\u79f0\u7684\u5143\u53cd\u53f7\u3002
\u6269\u5c55\u8d44\u6599\u51fd\u6570\u5728\u5176\u5b9a \u4e49\u57df\u7684\u67d0\u4e9b\u5c40\u90e8\u533a\u57df\u6240\u8fbe\u5230\u7684\u76f8\u5bf9 \u6700\u5927\u503c\u6216\u76f8\u5bf9\u6700\u5c0f\u503c\u3002\u5f53\u51fd\u6570\u5728\u5176 \u5b9a\u4e49\u57df\u7684\u67d0\u4e00\u70b9\u7684\u503c\u5927\u4e8e\u8be5\u70b9\u5468\u56f4 \u4efb\u4f55\u70b9\u7684\u503c\u65f6\uff0c\u79f0\u51fd\u6570\u5728\u8be5\u70b9\u6709\u6781 \u5927\u503c; \u5f53\u51fd\u6570\u5728\u5176\u5b9a\u4e49\u57df\u7684\u67d0\u4e00\u70b9\u7684\u503c\u5c0f\u4e8e\u8be5\u70b9\u5468\u56f4\u4efb\u4f55\u70b9\u7684\u503c\u65f6\uff0c \u79f0\u51fd\u6570\u5728\u8be5\u70b9\u6709\u6781\u5c0f\u503c\u3002\u8fd9\u91cc\u7684\u6781\u5927\u548c\u6781\u5c0f\u53ea\u5177\u6709\u5c40\u90e8\u610f\u4e49\u3002
\u56e0\u4e3a\u51fd \u6570\u7684\u4e00\u4e2a\u6781\u503c\u53ea\u662f\u5b83\u5728\u67d0\u4e00\u70b9\u9644\u8fd1 \u7684\u5c0f\u8303\u56f4\u5185\u7684\u6781\u5927\u503c\u6216\u6781\u5c0f\u503c\u3002\u51fd \u6570\u5728\u5176\u6574\u4e2a\u5b9a\u4e49\u57df\u5185\u53ef\u80fd\u6709\u8bb8\u591a\u6781 \u5927\u503c\u6216\u6781\u5c0f\u503c\uff0c\u800c\u4e14\u67d0\u4e2a\u6781\u5927\u503c\u4e0d \u4e00\u5b9a\u5927\u4e8e\u67d0\u4e2a\u6781\u5c0f\u503c\u3002
\u8bc1\u660e\uff1a\u6839\u636e\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u6027\u8d28\u6709\uff1a
AT=-A
|A|=|AT|=|-A|=\uff08-1\uff09n|A|=-|A|
\u7531\u4e8en\u4e3a\u5947\u6570
\u6240\u4ee5|A|=0
\u8bbeA\u4e3an\u7ef4\u65b9\u9635\uff0c\u82e5\u6709A'=-A\uff0c\u5219\u79f0\u77e9\u9635A\u4e3a\u53cd\u5bf9\u79f0\u77e9\u9635\u3002\u5bf9\u4e8e\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u5b83\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5168\u4e3a\u96f6\uff0c\u800c\u4f4d\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u4e24\u4fa7\u5bf9\u79f0\u7684\u5143\u53cd\u53f7\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u884c\u5217\u5f0f\u53ef\u4ee5\u770b\u505a\u662f\u6709\u5411\u9762\u79ef\u6216\u4f53\u79ef\u7684\u6982\u5ff5\u5728\u4e00\u822c\u7684\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\u7684\u63a8\u5e7f\u3002\u6216\u8005\u8bf4\uff0c\u5728 n \u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\uff0c\u884c\u5217\u5f0f\u63cf\u8ff0\u7684\u662f\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u5bf9\u201c\u4f53\u79ef\u201d\u6240\u9020\u6210\u7684\u5f71\u54cd\u3002
\u53cd\u5bf9\u79f0\u77e9\u9635\u5177\u6709\u5f88\u591a\u826f\u597d\u7684\u6027\u8d28\uff0c\u5982\u82e5A\u4e3a\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u5219A'\uff0c\u03bbA\u5747\u4e3a\u53cd\u5bf9\u79f0\u77e9\u9635\uff1b\u82e5A,B\u5747\u4e3a\u53cd\u5bf9\u79f0\u77e9\u9635\uff0c\u5219A\u00b1B\u4e5f\u4e3a\u53cd\u5bf9\u79f0\u77e9\u9635\uff1b\u8bbeA\u4e3a\u53cd\u5bf9\u79f0\u77e9\u9635\uff0cB\u4e3a\u5bf9\u79f0\u77e9\u9635\uff0c\u5219AB-BA\u4e3a\u5bf9\u79f0\u77e9\u9635\uff1b\u5947\u6570\u9636\u53cd\u5bf9\u79f0\u77e9\u9635\u7684\u884c\u5217\u5f0f\u5fc5\u4e3a0\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1\u2014\u2014\u53cd\u5bf9\u79f0\u77e9\u9635
简单计算一下即可，答案如图所示
证明：根据反对称矩阵的性质有：
A T =-A，
|A|=|A T |=|-A|=（-1） n |A|=-|A|
由于n为奇数，
所以|A|=0．
绛旓細濂囨暟闃跺弽瀵圭О鐭╅樀鐨勮鍒楀紡涓0銆傝瘉鏄庤繃绋嬶細璁続涓哄弽瀵圭О鐭╅樀锛屽嵆鏈 鏁呮湁锛氬綋n涓哄鏁版椂锛屽氨鐢 浜庢槸琛屽垪寮忕瓑浜0銆
绛旓細璇佹槑锛氭牴鎹鍙嶅绉扮煩闃鐨勬ц川鏈夛細AT=-A |A|=|AT|=|-A|=锛-1锛塶|A|=-|A| 鐢变簬n涓濂囨暟 鎵浠A|=0 璁続涓簄缁存柟闃碉紝鑻ユ湁A'=-A锛屽垯绉扮煩闃礎涓哄弽瀵圭О鐭╅樀銆傚浜庡弽瀵圭О鐭╅樀锛屽畠鐨勪富瀵硅绾夸笂鐨勫厓绱犲叏涓洪浂锛岃屼綅浜庝富瀵硅绾夸袱渚у绉扮殑鍏冨弽鍙枫
绛旓細璇佹槑锛氭牴鎹鍙嶅绉扮煩闃鐨勬ц川鏈夛細AT=-A |A|=|AT|=|-A|=锛-1锛塶|A|=-|A| 鐢变簬n涓濂囨暟 鎵浠A|=0 璁続涓簄缁存柟闃碉紝鑻ユ湁A'=-A锛屽垯绉扮煩闃礎涓哄弽瀵圭О鐭╅樀銆傚浜庡弽瀵圭О鐭╅樀锛屽畠鐨勪富瀵硅绾夸笂鐨勫厓绱犲叏涓洪浂锛岃屼綅浜庝富瀵硅绾夸袱渚у绉扮殑鍏冨弽鍙枫
绛旓細锛2锛夎A涓鍙嶅绉扮煩闃锛岃嫢A鐨勯樁鏁颁负濂囨暟锛屽垯A鐨琛屽垪寮涓0锛汚鐨勯樁鏁颁负鍋舵暟锛屽垯鏍规嵁鍏蜂綋鎯呭喌璁＄畻銆傚畾鐞嗗強鍏璇佹槑 瀹氱悊1 濂囨暟闃跺弽瀵圭О鐭╅樀鐨勮鍒楀紡蹇呬负0銆
绛旓細绠鍗曡绠椾竴涓嬪嵆鍙紝绛旀濡傚浘鎵绀
绛旓細姣忎竴琛屾彁鍑-1锛屾湁涓涓(-1)^n=-1, n涓濂囨暟 鍐嶈浆缃 璁板師琛屽垪寮忎负A锛岃浆缃鐨勮鍒楀紡涓篈'A=(-1)^n*A'=-A'=-A 鎵浠=0 璁続锛孊涓鍙嶅绉扮煩闃锛孉B涓嶄竴瀹氭槸鍙嶅绉扮煩闃点傝A涓哄弽瀵圭О鐭╅樀锛岃嫢A鐨勯樁鏁颁负濂囨暟锛屽垯A鐨勮鍒楀紡涓0锛汚鐨闃鏁颁负鍋舵暟锛屽垯鏍规嵁鍏蜂綋鎯呭喌璁＄畻銆
绛旓細绠鍗曡绠椾竴涓嬪嵆鍙紝绛旀濡傚浘鎵绀
绛旓細绠鍗曡绠椾竴涓嬪嵆鍙紝绛旀濡傚浘鎵绀
绛旓細璁 A 涓 n 闃锛坣 涓濂囨暟锛鍙嶅绉扮煩闃锛屽垯 A^T = -A锛屽洜姝 |A^T| = |-A|锛屼篃鍗 |A| = (-1)^n*|A| = -|A|锛屾墍浠 |A| = 0 銆
绛旓細姣忎竴琛屾彁鍑-1锛屾湁涓涓(-1)^n=-1, n涓濂囨暟 鍐嶈浆缃 璁板師琛屽垪寮忎负A锛岃浆缃鐨勮鍒楀紡涓篈'A=(-1)^n*A'=-A'=-A 鎵浠=0 璁続锛孊涓鍙嶅绉扮煩闃锛孉B涓嶄竴瀹氭槸鍙嶅绉扮煩闃点傝A涓哄弽瀵圭О鐭╅樀锛岃嫢A鐨勯樁鏁颁负濂囨暟锛屽垯A鐨勮鍒楀紡涓0锛汚鐨闃鏁颁负鍋舵暟锛屽垯鏍规嵁鍏蜂綋鎯呭喌璁＄畻銆
