设[tex=1.143x1.071]DFelGZAPNOqMgdbfKVoEHA==[/tex]表示[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]的附属方阵，证明：[tex=5.786x1.357]cRSSutUe8lxP7o+KrExJjIlQDv25D1qSOdQh99TznTk=[/tex]，其中[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex]也是[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵。

2022-06-08

设[tex=1.143x1.071]DFelGZAPNOqMgdbfKVoEHA==[/tex]表示[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]的附属方阵，证明：[tex=5.786x1.357]cRSSutUe8lxP7o+KrExJjIlQDv25D1qSOdQh99TznTk=[/tex]，其中[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex]也是[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵。

答案：

证法一：由于对任意方阵[tex=0.714x1.0]YiLkHgl7MlxE+QjUplQUKA==[/tex]，均有[tex=9.214x1.429]eovYZMsrXZyb74+wL+Uc/BHvC5HeDtfH9yjewxBKNrPRxg9hp8NssgCaVCHRimnT[/tex]故可得[tex=13.071x1.429]5pJYP87S+A6e7g4ftA0XM8yxu/o575rAgkgdrIxfmIZl5ms5ntra3TL+KMKI8fUOxvaHP1EHnEfl9kZwawQpfEo04PYqSupuXnG9MiRh0DU=[/tex][tex=9.071x1.357]QXIFHXgd1y49Cd34BHSX9eEMn0paVVxn7NEptHBlrkq4Wi1iOpfa+Gji+Ysbl9dX[/tex][tex=16.5x1.429]GErWO8NFHgmwvLdX0qN+OL+mKdvFq+HPEQG41xRcfwTH27+BopTFX96AbzlO2WSRonZ+aRX+q8wwHjavM+/kfCDi+uBn+sVn654Wgh6ORI4=[/tex][tex=13.0x1.357]p/ySVc4F7f+Iy8c9mheTVr70ZBzSm8zeAMVc59Bpl1p2euF4vLN3gjBndEvw2tVt[/tex]在上式中，用[tex=3.571x1.357]6rO8kmb/oFyZTaPrDM0jFA==[/tex]，[tex=3.571x1.357]OIY9aXp9uPQCAFLHPqRoLw==[/tex]分别代替[tex=0.786x1.0]kEam2pLJe4uAYVdcny2W5g==[/tex]，[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]，等式仍成立，即有[tex=18.929x1.571]n5CnxYdBtb1MFneM5OSY5x6mZ9WqNIZ4iov0QYN1DNBuJbyBoJYkvpUelQ0M3GNU3t8Sp0+NTWFC5Vm+GKUWthl4Y/s0eMRBlbgk8WcNjIGK6YWqV1pu1Z+mCZSjDbaXGaQwCHBKzo2yEDR0uyh2/Q==[/tex][tex=20.357x1.571]Ax+fYB0YMmxtSu/XHqv8HRcjnNSVxyu/OhmT4EaR2Chu9yP9zf4kcDNI66Ndf6lL/5gIRmJnx7NsINgVAs8uye7ojZ/F/lKRZziwffHOC6Tgl1sRvIQV8PPVabWUzp5SZ48isrq3f5odkU9x+6zNPHxwx81ArInmsHgBPTcYapQ=[/tex]总存在实数[tex=0.929x1.0]mQGdf3XTfQx0Qped0rrM9g==[/tex]，使得当[tex=2.857x1.071]1dLmz12Xg2FNPyOmTFOSCw==[/tex]时有[tex=6.071x1.5]n5CnxYdBtb1MFneM5OSY50X/z3WJC43CJBs6oAZUdL4=[/tex]，[tex=6.071x1.5]5NvvHO3nYvoZd4EjWHs/FgAvdOZlPUYOFlP6FM/YLTI=[/tex]，于是当[tex=2.857x1.071]1dLmz12Xg2FNPyOmTFOSCw==[/tex]时总有[tex=21.143x1.571]7Qs5pdoHbVQ80jP1w4aQrw0b9DecdE3OFl3aJPcyZ2dFPQjnG/gETEDKlxeJQLT+MS6oUKbEZg+acjTOoCAARmcMyUFOa1nasW1F5gbe4NLyGXX+xVC/pIxX7+I1qaaNycAn01a2syhtXOpfMUqBLcu+iqf/KfuQtYQ6IWEBX9k=[/tex]设左右两端第[tex=0.357x1.0]O88k7AtkDgTC9kv/8dY0lg==[/tex]行[tex=0.429x1.214]rmIPPJrP+tFN2kAYPlU/4g==[/tex]列的元素分别为[tex=2.429x1.357]NnryxVQG/4Slr8GwjSImmQ==[/tex]及 $g_{i j}(x)$, 上式表明[tex=5.571x1.357]Bu51vsOmkVMIwGZ82IG2i7eIOIpKo1BkCZqNthonI4c=[/tex]，[tex=6.5x1.214]ZRqTfVgPAL/6sRO3GmU0KeG6mCOoRm90Q0SYALxk6Zc=[/tex]。它们都是[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]的多项式，且对所有[tex=2.857x1.071]1dLmz12Xg2FNPyOmTFOSCw==[/tex]都相等，故两者恒等。 特别地，[tex=1.857x1.0]bOlCq/PHWhsSVMaVf7Obdg==[/tex]时也相等，从而得到[tex=5.786x1.357]Ak1mLy74GD7JjkvBP6IwFl14NkQ2SR0HW2LSUtBq36c=[/tex]。证法二：[tex=24.786x2.786]acb9w/eWm89xBAnv5bdfvFKAqMkYAselyxViamnyBUJ5gUZoLwYu4wuHEcPFwVtzzURav++wJ7b/WWYCpC41+IztifQfnKlkYja9ND7LNy189zDgFx4iLt5m87x4sHVP5maZPMNdcyZv7igSXjObexUa967di8Hvs6sudK/OzsioF5Q5vYha1yoIaSyGpwbnuR59u/PZ2MNj318jBb3ptQ==[/tex][tex=27.571x3.786]BT+ovw2kg6w7fKlDGSonH6XBA6SqNSW7MfSArkD305MROafj7ItM6aGylBumjll9F/t4+NpJHsR5bgMnPNTuLOCKz/aLC9qQiFtimoALHJmu5Vhad5xgSBKvFQGq8TLMamrg3clNCVFF+CF549ufh3ytwJ4L8R1g4thPp3Xd4BO1+4SrttzsuVy/1GeZeWvncUkDkDi07Qcd1WfJWVtRiBefKGtl88ylB1SCHXltfwagRxTpQHNiaUQ9XN27ywWnCBC6K5jvIMwtLzLCnbMunQ==[/tex][tex=16.857x2.786]15Pn+8NM5wH1mMbWvFBr7cDY53xa8pfaW3c8L1VpZxrs4VLwxaGLv4JY6TbkPOa70wGESIMqwUSF5Qyrm5Cj3/XPx3YvdMNjJVXTH5Uc6DQAY6R5NN6Vqcxj438EfmuSdLaUE5oV86wZwOVrY5c0CkElQ1tTmywZ6eQ5SmhftRoL/l/ZRlCgiw8MjUfAnka4[/tex][tex=36.5x3.643]c9l6PLP5LbbOQk0kDmV8foXA9BIDfZBKh6opkJUAaiqZLblGweDM2svMZGUdEbL4TIkF2e69Xi9k5FsUKL0ixNuE3yl6Jd7gc9Q1G1W8qWQg6nDQk2/veHytvzRxSkMVC0GNwt1z4fGI47IkbcOWm92znDa0izGkziXgE0KNjUXTNLxnl15zeJhUbNx7quq004lgswcoX+QMuqfcTVracvfI924ySV1YeB/C40bPW65bf25PYJPQs0mK0SckRgnkY8BYyNazsslQB3pTAz6+x5w2Iu2E4Zyx3pKweTeljg2qB7h6geXjrrAt4Kgb4kNd[/tex][tex=40.071x3.571]yfxO93wxHKIkBPRhsO5WS13KZusKnQd+xK4akOhSFdGUGmGvBQRm9Jy4KngMoyUicKnyD3BU/EswB3wnipp9oYjwWcPDzgnCHIRKk4CiOjRO7nPWggCTKGqp386/DO8k1AFXP/F98Ceuee//KetUpT5h0VK3deBOLMXzlirab10mJkvjQr6MInR1XyMkNKcMeo1qK1/n6chaPicA12yze9djL9wJL/Z5ESk++/AUweFUT8paCF/r48Np7sXT47/Wz8V+DViZ4TtYU6B+5ZmiDOW+PadXadTPo2V7Z4yJQ08Nu/EJsD9gEYerONxU1zCr3VYf9/qTUk48jbT3iwkl2zXQ6UOiwC3frNLBQ0fswnFISY6YmmwtecV5W6DBD7tI[/tex][tex=20.786x2.714]huqtSlXGruInNyozEqvaMYpCVtSIdtIvGUiuUjc7cxCL10RSiRsoFwlGHuD8WOEy2R0y6biHCnsT3AUKZaoDOa+dpJYrLGJZbpbregEkzQTxu/W7ibBCOsqrvSpz157O[/tex].

举一反三

内容

0
设[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]的每一行上都恰有2个元素为1，而其他元素为零，[tex=0.571x1.0]EnSTrJsHc9I00M+IaN7q+w==[/tex]是元素全为1的[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵。求出所有适合[tex=5.0x1.357]4+sHTHuBuyOCEg+k8CDzeQ==[/tex]的[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]。
1
设 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶方阵 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 满足 [tex=2.714x1.214]+ZPJntj7xYfllBYE3zVGBw==[/tex]，证明（1）[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 可逆；（2）[tex=9.786x1.357]06AJfdzBDu7SdZ9anbGLIPmuCvp8KJZXpIhBloDxMHk=[/tex] .
2
若 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]为[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶方阵，[tex=1.143x1.071]LQkrj7qu81kJB+TsdvGtMw==[/tex]为 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]的伴随矩阵，证明:[tex=12.786x4.5]uyNdoq9cJoDKbikMVGY4jW1yMSrW1BNdK/VdPCCOABZcjMDLRGVVlBO9nQ5XhIOlnDPa3If7B91qAfkFGrnsX7jBYDjogHb+BMulTtfLojcKdF93IUITPbznYf3i5bjTKPawEEunvGNLctRnD2VnEQ==[/tex]
3
设[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶可逆方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]的伴随矩阵为[tex=1.143x1.071]F0wJ6Hm8K7uRqU9zt3sS4A==[/tex]，[tex=5.714x1.5]7+UslwtIbOlbpBz5l2fvMS8pAL2LPmbb1oXRYXwsx+g=[/tex] .
4
适合[tex=2.786x1.214]5YJ7IJv26przrQ/Z5urQMQ==[/tex]的方阵称为对称方阵。证明：[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶对称方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]和[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex]的乘积[tex=1.571x1.0]mCjAngcIqtveplNftuY0BQ==[/tex]为对称方阵的充分必要条件是，方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]和[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex]可交换。