设 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶矩阵[tex=3.571x1.357]7K89EAiqbgRkVf5frr2x25+2ay1ha16/s2MrqtRX+/U=[/tex] 的行列式等于零, 证明: [tex=1.143x1.071]dlHppezehhhJt6WmQH9aoA==[/tex] 的秩不超过 1.
举一反三
- 求证: [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶矩阵 [tex=3.571x1.357]7K89EAiqbgRkVf5frr2x25+2ay1ha16/s2MrqtRX+/U=[/tex] 是正定阵, 其中 [tex=5.929x2.643]FeWLmF3VqZSfVJC/QokCEn4mIi8d5os7+du0n1ggFpfRcjGRpWTVy06gumtSUyK8[/tex] 是任意的正整数.
- 设 [tex=3.571x1.357]7K89EAiqbgRkVf5frr2x25+2ay1ha16/s2MrqtRX+/U=[/tex] 为 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶方阵, 定义函数 [tex=6.5x2.857]yI0/YP6f+1zdqtC3LVgdQx5kfjBFfACdIlTZbt+nvkmp6t9yq8iuvsaN4P780Vx0[/tex], 设 [tex=0.643x1.0]WUJ/JHItsc3Bqx1WYNJcrg==[/tex] 为 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶可逆矩阵, 使得对任 意的 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶方阵 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 成立: [tex=7.643x1.571]UYfuvlBjvyBtnTMT/LcB68AO+Z9we4bTHzeszC2uNjY=[/tex], 证明: 存在非零常数 [tex=0.5x0.786]hycNLgozeED/VkKdun7zdA==[/tex], 使得 [tex=4.0x1.357]9H+B938/Tud1uW1GYTpYFhCan8KCnZCA7uSC8i04frA=[/tex]
- 证明, [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶实对称矩阵[tex=3.571x1.357]7K89EAiqbgRkVf5frr2x25+2ay1ha16/s2MrqtRX+/U=[/tex]是正定的，必要且只要对于任意[tex=8.929x1.143]d81iwVDPCo2Pvt1l0Mp5tqgyiGep7S9S9dNH84YBmb0nqz+78AiOMxynWiGwGxph[/tex]，[tex=0.571x1.0]CQkpoDeAAI+5FKIfe1wVCA==[/tex]阶子式[tex=20.071x4.5]TIwZYBkNsy31H1RNd/OloB0w8VG7gvV5wru2KiA0NQ3Sb9S/muuh23fLjA2oE4sBA6IBT9ZsVxS6GnNI0yyEFUjMuayi+5ujIgVhwhVi4bionz79/ALTbwkNOFfxy41W/E3hqpOwbvhDucVImS7VvEBAI8m1VIDjQZIpFU3fIkwpz1clplRHDtGwXnXthAfNmkf5+keaYkeTle7s+PqdEv/jbzI74v9DhySfD04oriqlhAYGFNFvLvge7cxM3+079ShOKtFpGwtydqkly9wndKbDTmw+8Kjj96125weynB1x4K3tCxSOWt4RIE4jIfumSJIO4pm1kd3OfNr3uLEEAEUns5uWppLcy9y89fbmtZVg0SfWAKo8L3BdkpO4fcPeJj9ijjFE7bZ5g03cN9tSUg==[/tex]
- 证明, [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶实对称矩阵[tex=3.571x1.357]7K89EAiqbgRkVf5frr2x25+2ay1ha16/s2MrqtRX+/U=[/tex]是正定的，必要且只要对于任意[tex=8.929x1.143]d81iwVDPCo2Pvt1l0Mp5tqgyiGep7S9S9dNH84YBmb0nqz+78AiOMxynWiGwGxph[/tex]，[tex=0.571x1.0]CQkpoDeAAI+5FKIfe1wVCA==[/tex]阶子式[tex=20.071x4.5]TIwZYBkNsy31H1RNd/OloB0w8VG7gvV5wru2KiA0NQ3Sb9S/muuh23fLjA2oE4sBA6IBT9ZsVxS6GnNI0yyEFUjMuayi+5ujIgVhwhVi4bionz79/ALTbwkNOFfxy41W/E3hqpOwbvhDucVImS7VvEBAI8m1VIDjQZIpFU3fIkwpz1clplRHDtGwXnXthAfNmkf5+keaYkeTle7s+PqdEv/jbzI74v9DhySfD04oriqlhAYGFNFvLvge7cxM3+079ShOKtFpGwtydqkly9wndKbDTmw+8Kjj96125weynB1x4K3tCxSOWt4RIE4jIfumSJIO4pm1kd3OfNr3uLEEAEUns5uWppLcy9y89fbmtZVg0SfWAKo8L3BdkpO4fcPeJj9ijjFE7bZ5g03cN9tSUg==[/tex]
- 设 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 为 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶矩阵，证明 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 为正交阵的充要条件是 [tex=1.143x1.071]dlHppezehhhJt6WmQH9aoA==[/tex] 为正交阵.