线性代数特殊四阶行列式求解 线性代数，四阶行列式方程求解
\u7ebf\u6027\u4ee3\u65704\u9636\u884c\u5217\u5f0f\u600e\u4e48\u8ba1\u7b97\uff1f\u884c\u5217\u5f0f\u7684\u503c=\u6240\u6709\u6765\u81ea\u4e0d\u540c\u884c\u4e0d\u540c\u5217\u7684\u5143\u7d20\u7684\u4e58\u79ef\u7684\u548c\u3002\u6bcf\u4e00\u9879\u90fd\u662f\u4e0d\u540c\u884c\u4e0d\u540c\u5217\u5143\u7d20\u7684\u4e58\u79ef\u3002\u56e0\u4e3aa11\u548ca23\u5360\u7528\u4e861\uff0c2\u884c\u548c1\uff0c3\u5217\uff0c\u6240\u4ee5\u5269\u4e0b\u7684\u4e24\u4e2a\u5143\u7d20\u6765\u81ea3\uff0c4\u884c\u76842\uff0c4\u5217\uff1b 1\u3001\u7b2c\u4e09\u884c\u53d6\u7b2c\u4e8c\u5217\uff0c\u5373a32\uff0c\u5219\u7b2c\u56db\u884c\u53ea\u80fd\u53d6\u7b2c\u56db\u5217\uff0c\u5373a44\uff0c\u4e5f\u5c31\u662fa11a23a32a44; 2\u3001\u7b2c\u4e09\u884c\u53d6\u7b2c\u56db\u5217\uff0c\u5373a34\uff0c\u5219\u7b2c\u56db\u884c\u53ea\u80fd\u53d6\u7b2c\u4e8c\u5217\uff0c\u5373a42\uff0c\u4e5f\u5c31\u662fa11a23a34a42; 3\u3001\u6bcf\u4e00\u9879\u7684\u6b63\u8d1f\u53f7\u53d6\u51b3\u4e8e\u9006\u5e8f\u6570\uff0c\u5bf9\u4e8ea11a23a32a44\uff0c\u9006\u5e8f\u6570\u53d6\u51b3\u4e8e\u30101 3 2 4\u3011\uff0c\u9006\u5e8f\u6570\u4e3a1\uff0c\u6240\u4ee5\u53d6\u8d1f\u53f7 4\u3001\u5bf9\u4e8ea11a23a34a42\uff0c\u9006\u5e8f\u6570\u53d6\u51b3\u4e8e\u30101 3 4 2\u3011\uff0c\u9006\u5e8f\u6570\u4e3a2\uff0c\u6240\u4ee5\u53d6\u6b63\u53f7
\u56db\u9636\u884c\u5217\u5f0f\u8981\u6bd4\u4e09\u9636\u884c\u5217\u5f0f\u590d\u6742\u5f97\u591a\uff0c\u662f\u771f\u6b63\u610f\u4e49\u7684\u9ad8\u9636\u884c\u5217\u5f0f\u3002\u6c42\u56db\u9636\u884c\u5217\u5f0f\u7684\u65b9\u6cd5\u6709\u5f88\u591a\uff0c\u53ef\u4ee5\u76f4\u63a5\u7528\u5c55\u5f00\u516c\u5f0f\uff1b\u4e5f\u53ef\u4ee5\u5316\u56db\u9636\u884c\u5217\u5f0f\u4e3a\u4e0a\u4e09\u89d2\u884c\u5217\u5f0f\uff1b\u53ef\u4ee5\u628a\u884c\u5217\u5f0f\u67d0\u884c\u6216\u8005\u5217\u5c3d\u53ef\u80fd\u7684\u591a\u5316\u51fa\u96f6\uff0c\u7136\u540e\u6309\u8fd9\u4e00\u884c\u6216\u5217\u5c55\u5f00\u3002
\u8fd9\u91cc\u53cd\u590d\u7528\u5230\u4e86\u51e0\u4e2a\u6027\u8d28\uff1a\u884c\u5217\u5f0f\u7684\u503c\u7b49\u4e8e\u884c\u5217\u5f0f\u67d0\u4e00\u884c(\u6216\u5217)\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\u52a0\u5230\u53e6\u4e00\u884c(\u53e6\u4e00\u5217)\u4e0a\uff1b\u4ea4\u6362\u884c\u5217\u5f0f\u7684\u67d0\u4e24\u884c(\u6216\u5217)\u884c\u5217\u5f0f\u524d\u9762\u8981\u4e58\u4e00\u4e2a\u8d1f\u53f7\uff1b\u884c\u5217\u5f0f\u67d0\u884c\u6216\u8005\u5217\u6709\u516c\u56e0\u5b50\uff0c\u53ef\u4ee5\u628a\u8fd9\u4e2a\u516c\u56e0\u5b50\u63d0\u51fa\u53bb\u3002
\u884c\u5217\u5f0f\u4e2d\u6700\u7b80\u5355\u7684\u662f\u5bf9\u89d2\u884c\u5217\u5f0f\uff0c\u4e0a\u4e09\u89d2\u884c\u5217\u5f0f\uff0c\u4e0b\u4e09\u89d2\u884c\u5217\u5f0f\uff0c\u8fd9\u4e09\u7c7b\u884c\u5217\u7c7b\u7684\u503c\u5747\u4e3a\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e4b\u548c\u3002\u5982\u4f55\u628a\u4efb\u4e00\u4e2a\u884c\u5217\u5f0f\u8f6c\u5316\u4e3a\u8fd9\u7c7b\u884c\u5217\u5f0f\u662f\u9700\u8981\u540c\u5b66\u52a8\u8111\u7684\u5730\u65b9\u3002
\u56db\u9636\u884c\u5217\u5f0f\u8981\u6bd4\u4e09\u9636\u884c\u5217\u5f0f\u590d\u6742\u5f97\u591a\uff0c\u662f\u771f\u6b63\u610f\u4e49\u7684\u9ad8\u9636\u884c\u5217\u5f0f\u3002\u6c42\u56db\u9636\u884c\u5217\u5f0f\u7684\u65b9\u6cd5\u6709\u5f88\u591a\uff0c\u53ef\u4ee5\u76f4\u63a5\u7528\u5c55\u5f00\u516c\u5f0f\uff1b\u4e5f\u53ef\u4ee5\u5316\u56db\u9636\u884c\u5217\u5f0f\u4e3a\u4e0a\u4e09\u89d2\u884c\u5217\u5f0f\uff1b\u53ef\u4ee5\u628a\u884c\u5217\u5f0f\u67d0\u884c\u6216\u8005\u5217\u5c3d\u53ef\u80fd\u7684\u591a\u5316\u51fa\u96f6\uff0c\u7136\u540e\u6309\u8fd9\u4e00\u884c\u6216\u5217\u5c55\u5f00\u3002
\u8fd9\u91cc\u53cd\u590d\u7528\u5230\u4e86\u51e0\u4e2a\u6027\u8d28\uff1a\u884c\u5217\u5f0f\u7684\u503c\u7b49\u4e8e\u884c\u5217\u5f0f\u67d0\u4e00\u884c(\u6216\u5217)\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\u52a0\u5230\u53e6\u4e00\u884c(\u53e6\u4e00\u5217)\u4e0a\uff1b\u4ea4\u6362\u884c\u5217\u5f0f\u7684\u67d0\u4e24\u884c(\u6216\u5217)\u884c\u5217\u5f0f\u524d\u9762\u8981\u4e58\u4e00\u4e2a\u8d1f\u53f7\uff1b\u884c\u5217\u5f0f\u67d0\u884c\u6216\u8005\u5217\u6709\u516c\u56e0\u5b50\uff0c\u53ef\u4ee5\u628a\u8fd9\u4e2a\u516c\u56e0\u5b50\u63d0\u51fa\u53bb\u3002\u662f\u4e3a\u4e4b\u540e\u5b66\u4e60\u7684\u77e9\u9635\u505a\u51c6\u5907\uff0c\u5728\u77e9\u9635\u7684\u5b66\u4e60\u4e2d\uff0c\u4e00\u4e2a\u91cd\u8981\u7684\u6280\u5de7\u662f\u5b9e\u884c\u53d8\u6362\u3002
\u884c\u5217\u5f0f\u7b80\u4ecb\uff1a
\u884c\u5217\u5f0f\uff08determinant\uff09\u5728\u6570\u5b66\u4e2d\uff0c\u662f\u4e00\u4e2a\u51fd\u6570\uff0c\u5176\u5b9a\u4e49\u57df\u4e3adet\u7684\u77e9\u9635A\uff0c\u53d6\u503c\u4e3a\u4e00\u4e2a\u6807\u91cf\uff0c\u5199\u4f5cdet(A)\u6216 |A|\u3002\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
| 1 0 0 2 |
| 0 1 2 0 |
| 0 3 4 0 |
| 3 0 0 4 |
= det([ 1 2 ; 3 4 ]ᵀ) ×det([ 1 2 ; 3 4 ]ᵀ)
= (-2)×(-2)=4
5. 行列式计算如下：
| x a a a |
| a x a a |
| a a x a |
| a a a x |
将第二、三、四列分别加到第一列，得：
| x+3a a a a |
| x+3a x a a |
| x+3a a x a |
| x+3a a a x |
令第二、三、四行分别减去第一行，得：
| x+3a a a a |
| 0 x-a 0 0 |
| 0 0 x-a 0 |
| 0 0 0 x-a |
化为上三角型，则行列式为主对角线元素相乘：(x+3a)(x-a)³
6. 行列式计算如下：
| 1 2 -1 1 |
| 3 6 1 2 |
| 1 -1 2 1 |
| 1 0 3 -2 |
第二行减去第一行的3倍，第三、四行分别减去第一行，得：
| 1 2 -1 1 |
| 0 0 4 -1 |
| 0 -3 3 0 |
| 0 -2 4 -3 |
按第一列展开，得：
| 0 4 -1 |
| -3 3 0 |
| -2 4 -3 |
第二行减去第三行的3/2倍，得：
| 0 4 -1 |
| 0 -3 9/2 |
| -2 4 -3 |
按第一列展开，得：
-2×(4×9/2-3)=-30
绛旓細-2脳(4脳9/2-3)=-30
绛旓細鍥涢樁琛屽垪寮忚姣斾笁闃惰鍒楀紡澶嶆潅寰楀锛屾槸鐪熸鎰忎箟鐨勯珮闃惰鍒楀紡銆姹傚洓闃惰鍒楀紡鐨勬柟娉曟湁寰堝锛屽彲浠ョ洿鎺ョ敤灞曞紑鍏紡锛涗篃鍙互鍖栧洓闃惰鍒楀紡涓轰笂涓夎琛屽垪寮忥紱鍙互鎶婅鍒楀紡鏌愯鎴栬呭垪灏藉彲鑳界殑澶氬寲鍑洪浂锛岀劧鍚庢寜杩欎竴琛屾垨鍒楀睍寮銆傝繖閲屽弽澶嶇敤鍒颁簡鍑犱釜鎬ц川锛氳鍒楀紡鐨勫肩瓑浜庤鍒楀紡鏌愪竴琛(鎴栧垪)涔樹互涓涓父鏁板姞鍒板彟涓琛(...
绛旓細1銆佽В娉曚竴锛氱涓琛岀涓涓暟涔樹互瀹冪殑浠ｆ暟浣欏瓙寮忥紝鍔犵涓琛岀浜屼釜鏁颁箻璐熶竴涔樺畠鐨勪唬鏁颁綑瀛愬紡锛屽姞涓婄涓琛岀涓変釜鏁颁箻浠ｆ暟浣欏瓙寮忥紝鍔犱笂绗竴琛岀鍥涗釜鏁颁箻璐熶竴涔樺畠鐨勪唬鏁颁綑瀛愬紡銆2銆佽В娉曚簩锛氬皢鍥涢樁琛屽垪寮忓寲鎴愪笂涓夎琛屽垪寮忥紝鐒跺悗涔樹互瀵硅绾夸笂鐨勫洓涓暟銆傛樉鐒剁浜屽垪鏈夊緢澶0锛屾墍浠ュ皢绗簲琛屽噺鍘荤浜岃...
绛旓細琛屽垪寮=|(1,2,2,2)(1,0,0,0)(3,-1,-4,-6)(1,2,-1,-7)| 銆恈3-c1銆乧4-c1*2銆=2*|(1,1,1)(1,4,6)(2,-1,-7)| 銆愭寜r2灞曞紑锛屾彁鍑簉1銆乺2鐨勫叕鍥犲瓙銆=2*|(1,0,0)(1,3,5)(2,-3,-9)| 銆恈2-c1銆乧3-c1銆=2*|(3,5)(-3,-9)| 銆愭寜r1灞曞紑銆...
绛旓細鏂规硶涓锛氱涓琛屼箻锛-1锛夊姞鍒扮浜岃锛岀涓琛屼箻锛-2锛夊姞鍒扮涓夎锛岀涓琛屼箻锛-3锛夊姞鍒扮鍥涜锛岃繖鏍风涓鍒楀氨鍙湁涓涓潪闆跺厓绱犱簡銆傚啀鐢ㄥ睍寮瀹氱悊锛屾寜绗竴鍒楀睍寮銆傛柟娉曚簩锛氱浜岃鍔犲埌绗竴琛岋紝绗笁琛屽姞鍒扮涓琛岋紝绗洓琛屽姞鍒扮涓琛岋紝杩欐牱绗竴琛岃兘澶熸彁鍑哄叕鍥犲瓙10,绗竴琛屽氨鍖栦负閮芥槸1鐨勫厓绱犮
绛旓細r2-r1*2銆乺3-r1銆乺4-r1*2 D4=|1 2 1 2| 0 -3 -1 -3 0 -1 0 0 0 -3 0 -3 鎻愬嚭r2鍏洜瀛(-1)锛涗氦鎹2銆乧3 =|1 1 2 2| 0 1 3 3 0 0 -1 0 0 0 -3 -3 浜ゆ崲r3銆乺4锛涗氦鎹3銆乧4 =|1 ...
绛旓細鍥涢樁琛屽垪寮鏄竴涓敱鍥涗釜琛屾垨鍥涗釜鍒楃粍鎴愮殑鏂归樀锛屾瘡涓厓绱犻兘鏈変竴涓壒瀹氱殑浣嶇疆锛岀敱琛屽彿鍜屽垪鍙风殑缁勫悎纭畾銆傚畠涓庝簩闃惰鍒楀紡鍜屼笁闃惰鍒楀紡涓鏍凤紝閮芥槸鐢ㄤ簬瑙ｅ喅绾挎т唬鏁涓悇绉嶉棶棰樼殑閲嶈宸ュ叿銆傚洓闃惰鍒楀紡鐨勬瘡涓涓厓绱犻兘鏄竴涓暟瀛楋紝杩欎簺鏁板瓧閫氳繃鐗瑰畾鐨勮绠楄鍒欏彲浠ュ緱鍒颁竴涓粨鏋溿傚瑙掔嚎娉曞垯璁＄畻 鍥涢樁琛屽垪寮忔渶甯哥敤...
绛旓細浣犲ソ锛佸厛鎶婄浜屼笁鍥涘垪閮藉姞鍒扮涓鍒椾笂锛屽啀鎶婄涓琛屼箻-1鍔犲埌绗簩涓夊洓琛屼笂锛屽氨鍖栨垚浜嗕笂涓夎琛屽垪寮锛岀瓟妗堟槸17*(-3)*(-3)*(-3)=-459銆傜粡娴庢暟瀛﹀洟闃熷府浣犺В绛旓紝璇峰強鏃堕噰绾炽傝阿璋紒
绛旓細3. 鑼冨痉钂欏痉琛屽垪寮忕殑閫熻В鍏紡鑼冨痉钂欏痉琛屽垪寮忕殑鐗规畩涔嬪鍦ㄤ簬鍏跺浐瀹氱殑鍏紡锛岄潰瀵硅繖绫昏鍒楀紡锛岀洿鎺ュ鐢ㄥ叕寮 [(d-c)(d-b)(d-a)(c-b)(c-a)(b-a)]鍗冲彲锛屾棤闇绻佺悙璁＄畻銆傚浜庢洿楂橀樁鐨勭被浼兼儏鍐碉紝鑼冨痉钂欏痉宸茬粰鍑鸿В绛旀ā鏉裤4. 鍒涢犻浂鍏冪礌鐨勯檷闃剁瓥鐣ラ潰瀵归珮闃惰鍒楀紡锛屽埄鐢ㄨ鍒楀紡鐨勬ц川锛岄氳繃琛岋紙鍒楋級...
绛旓細瑙ｆ硶涓:灏嗙涓琛岀涓涓暟涔樹互瀹冪殑浠ｆ暟浣欏瓙寮忥紝鍔犵涓琛岀浜屼釜鏁颁箻璐熶竴涔樺畠鐨勪唬鏁颁綑瀛愬紡锛屽姞涓婄涓琛岀涓変釜鏁颁箻浠ｆ暟浣欏瓙寮忥紝鍔犱笂绗竴琛岀鍥涗釜鏁颁箻璐熶竴涔樺畠鐨勪唬鏁颁綑瀛愬紡;瑙ｆ硶浜:灏鍥涢樁琛屽垪寮鍖栨垚涓婁笁瑙掕鍒楀紡锛岀劧鍚庝箻浠ュ瑙掔嚎涓婄殑鍥涗釜鏁般傚皢鍥涢樁琛屽垪寮忓寲鎴愪笂涓夎琛屽垪寮忥紝鐒跺悗涔樹互瀵硅绾夸笂鐨勫洓...
