3阶行列式的展开式 三阶行列式的展开式是怎么计算的？n阶的呢？
\u4e09\u9636\u884c\u5217\u5f0f(3\u9636\u77e9\u9635)\u7684\u5c55\u5f00\u5f0f\u5b50\u66f2\u9762\u4e0a\u4efb\u4e00\u70b9(x,y,z)\u6ee1\u8db3x^2+y^2+z^2=R^2\uff0c\u56e0\u6b64\u88ab\u79ef\u51fd\u6570x^2+y^2+z^2=R^2\uff0c
\u8fdb\u800c\uff1a\u222b\u222bx^2+y^2+z^2ds = \u222b\u222bR^2ds = R^2*\u222b\u222bds = R^2 * 4\u03c0R^2 = 4\u03c0R^4
\u5176\u4e2d\u222b\u222bds\u8868\u793a\u7403\u9762x^2+y^2+z^2=R^2\u7684\u8868\u9762\u79ef\u53734\u03c0R^2
\u54c8\u2026\u6211\u521a\u597d\u5728\u590d\u4e60\u3002\u4e0d\u8fc7\uff0c\u8be5\u600e\u4e48\u8bf4\u5462\uff1f\u90a3\u4e2a\u2026\u4f60\u77e5\u9053n\u9636\u884c\u5217\u5f0f\u7684\u7ed3\u6784\u5417\uff1f\u5b83\u7684\u503c\u7b49\u4e8e\u6240\u6709\u53d6\u81ea\u4e0d\u540c\u884c\u4e0d\u540c\u5217\u7684n\u4e2a\u5143\u7d20\u7684\u4e58\u79ef\u7684\u4ee3\u6570\u548c\u3002\u800c\u6bcf\u4e00\u9879\u4e58\u79ef\u7684\u6b63\u8d1f\u53d6\u51b3\u4e8e\u5b83\u662f\u5076\u6392\u5217\uff08\u4e3a\u6b63\uff09\u8fd8\u662f\u5947\u6392\u5217\uff08\u4e3a\u8d1f\uff09\u2026\u81f3\u4e8e\u6392\u5217\u2026\u5443\uff0c\u4f60\u77e5\u9053\u5417\uff1f\u5509\u2026\u6211\u90fd\u4e0d\u77e5\u9053\u5728\u624b\u673a\u4e0a\u8be5\u600e\u4e48\u8868\u8fbe\u2026\u5982\u679c\u662f\u4e09\u9636\u884c\u5217\u5f0f\uff08\u4e0b\u6807\u6807\u5f97\u89c4\u8303\u7684\u8bdd\uff09\uff0c\u7b49\u4e8ea11a22a33\uff0ba12a23a31\uff0ba13a21a32\uff0da13a22a31\uff0da12a21a33\uff0da11a23a32. \u545c\u545c\u545c\u2026\u6211\u5feb\u8981\u54ed\u4e86\uff01\u987a\u4fbf\u8bf4\u4e00\u4e0b\uff1a\u201c\u6211\u662f\u5b66\u6570\u5b66\u7684\uff01\u201d\u6211\u89c9\u5f97\u4f60\u8fd8\u662f\u627e\u672c\u4e66\u5427\uff5e
三阶行列式展开如下所示:
绛旓細鎸夈琛屽垪寮忓睍寮瀹氱悊銆嬶紙鎷夋皬瀹氱悊锛夛紝鎶婅鍒楀紡鎸夋煇涓琛岋紙鎴栨煇涓鍒楋級灞曞紑锛屽嵆鍙妸涓涓涓夐樁琛屽垪寮鍖栦负涓変釜浜岄樁琛屽垪寮忋傚锛殀(a11,a12,a13)(a21,a22,a23)(a31,a32,a33)| 銆愭寜绗竴琛屽睍寮銆=a11*|(a22,a23)(a32,a33)| - a12*|(a21,a23)(a31,a33)| + a13*|(a21,a22)(a31,a32)| ...
绛旓細1 0 0 *(-1) 鎸夌3琛灞曞紑 =0
绛旓細涓夐樁琛屽垪寮忓睍寮濡備笅鎵绀:
绛旓細鍏跺搴旂殑涓夐樁琛屽垪寮忕殑灞曞紑寮忎负锛|A| = aei + bfg + cdh - ceg - bdi - afh 涓夎褰㈡硶锛圫arrus娉曞垯锛夋槸鍙︿竴绉嶇敤浜庤绠椾笁闃惰鍒楀紡鐨勬柟娉曪紝鍏舵楠ゅ涓嬶細1. 灏3x3鐭╅樀澶嶅埗涓ゆ骞跺皢鍏剁矘璐村埌鍙充晶锛屽舰鎴愪簡涓涓6x3鐨勭煩闃点2. 浠庡乏涓婅鍒板彸涓嬭鐨勫瑙掔嚎涓婄殑鏁扮浉涔樺苟姹傚拰锛屽嵆涓轰笁闃惰鍒楀紡鐨勭粨鏋溿...
绛旓細涓夐樁琛屽垪寮忓彲鐢ㄥ瑙掔嚎娉曞垯锛D = a11a22a33 + a12a23a31 + a13a21a32- a13a22a31 - a12a21a33 - a11a23a32銆倈a11 a12 a13|=a11a22a33-a11a23a32+a12a23a31-a12a21a33+a13a32a21-a13a22a31锛宎21 a22 a23銆俛31 a32 a33锛=a11a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31...
绛旓細鎸夌涓琛屾妸琛屽垪寮忓睍寮锛=a11*(a22*a33-a23*a32)-a12*(a21*33-a31*a23)+a13*(a21*a32-a31*a22)=(2-x)(x^2-x-4)+4x =-x^3+3x^2+6x-8 =(-x^3+4x^2)-(x^2-6x+8)(鎷嗛」锛屽垎缁勫垎瑙ｅ洜寮忥級=-x^2(x-4)-(x-2)(x-4)=-(x-4)(x-1)(x+2)瑕佽鍒楀紡涓洪浂 鍒檟=4鎴...
绛旓細3銆涓夐樁鐭╅樀琛屽垪寮忕殑灞曞紑璁＄畻鏂规硶锛氬綋涓涓煩闃电殑闃舵暟涓3鏃讹紝鍏惰鍒楀紡涓鑸敤灞曞紑鐨勬柟娉曟潵璁＄畻銆傚睍寮鏄妸涓鍏冧簩娆℃柟绋嬭〃杈惧紡鍒嗚В鎴愬涓畾涔夊悓绛夌殑涓鍏冧簩娆″紡銆備簩銆佷笁闃剁煩闃佃鍒楀紡璁＄畻姝ラ濡備笅锛氾紙1锛夐夊彇鐭╅樀涓竴琛屾垨涓鍒楋紝骞跺啓鍑虹煩闃琛屽垪寮忕殑灞曞紑寮锛涳紙2锛夋妸閫夊嚭鐨勮鎴栧垪鎹㈡垚涓庡叾浠栬锛堝垪锛変笉鍚岀殑鍏朵粬...
绛旓細鐢涓夐樁琛屽垪寮忕殑灞曞紑寮(12-4) 鍙婁唬鏁颁綑瀛愬紡锛屽皢涓夐樁琛屽垪寮廌鍙〃绀轰负D= a21A21 + a22A22 + a23A23锛屾寮忕О涓鸿鍒楀紡鎸夌浜岃鐨勫睍寮寮忋傚悓鏍凤紝琛屽垪寮忎篃鍙寜鍏朵粬琛屾垨鍒楀睍寮锛屼簬鏄瘡涓鍒楀紡鍙互琛ㄦ垚瀹冪殑鏌愪竴琛(鎴栨煇涓鍒)鐨勬瘡涓厓绱犱笌瀹冨搴斿厓绱犵殑浠ｆ暟浣欏瓙寮忎箻绉殑鍜岋紝鍗 D= ai1Ai1+ ai2Ai2...
绛旓細鏇查潰涓婁换涓鐐(x,y,z)婊¤冻x^2+y^2+z^2=R^2锛屽洜姝よ绉嚱鏁皒^2+y^2+z^2=R^2锛岃繘鑰岋細鈭埆x^2+y^2+z^2ds = 鈭埆R^2ds = R^2*鈭埆ds = R^2 * 4蟺R^2 = 4蟺R^4 鍏朵腑鈭埆ds琛ㄧず鐞冮潰x^2+y^2+z^2=R^2鐨勮〃闈㈢Н鍗4蟺R^2 ...
绛旓細1銆佸瑙掔嚎灞曞紑锛殀a1 b1| =a1b2-a2b1|a2 b2| |a1 b1 c1||a2 b2 c2|=a1b2c3+b1c2a3+c1a2b3-a3b2c1-b3c2a1-c3a2b1|a3 b3 c3|闄闃跺睍寮锛堥傚悎楂橀樁琛屽垪寮忥級濡涓夐樁琛屽垪寮 鎸夌涓闃跺睍寮|a b c||d e f |=a脳|e f|-b脳|d f|+c脳|d e| |g h i | |h i| |g i| ...
