设三阶矩阵[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]满足[tex=6.5x1.286]fKMuxXvMmkhZ6KBFJApU0RzCv0uzUPjL4nc3K7Aig7k=[/tex]，证明：[tex=3.214x1.286]CxRh2MuWfyX9bh9PvBg47Q==[/tex]可逆 .
举一反三
- 已知[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]为三阶对称矩阵且满足[tex=7.571x1.286]UftoPCbY1vsscBWCv7WWdWTmPBSQuX1bXssS0ZKFAo0=[/tex]；其中[tex=0.786x1.286]YggwMQ4w3PxfhkmL0NfgdQ==[/tex]为三阶单位矩阵 . 证明；（1）矩阵[tex=3.214x1.286]CxRh2MuWfyX9bh9PvBg47Q==[/tex]可逆，并求出[tex=4.929x1.286]UDCzTeZjplcUwcMi2zGvIfGEmkN+NtWm5jFlVbdJ7f4=[/tex]；（2）若矩阵[tex=8.214x3.643]GGH5qazUqq7NUCYOcK6l4MoNKp6iMwMC6U8h1ew2xuMXF623jmjDH3Y+PeCZFHHxRawlm4maS6EPdXiK7Ep6RuByIf4EgILr6Js2A3n62/V5Gmr0XFZVgg/3HWGihKA9[/tex]，求矩阵[tex=0.786x1.286]pi/GsQ3apuRt43V3XQq/tA==[/tex] .
- 同阶实对称矩阵[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]合同的充分必要条件是[input=type:blank,size:6][/input] . 未知类型：{'options': ['[tex=5.786x1.286]g8SwOoc3mjjpDvp+UxwKlmKIyJYBt/w89p9ioRDdWtw=[/tex]', '[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]为同型矩阵', '[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]的秩与正惯性指数都对应相等', '[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]的正惯性指数相等'], 'type': 102}
- 设[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]均为正定矩阵，证明[tex=5.0x2.786]075gCzZzsMRb6HYXYk9X92zk8W4u1qJBIO8aFf+ZsZxwp/haXQ2S0bij0nON3lddoX4sG6nvdaxHgFoCKqduPw==[/tex]为正定矩阵 .
- 以下命题是正确的是（ ），且说明理由：（1）对任何矩阵[tex=0.786x1.286]pi/GsQ3apuRt43V3XQq/tA==[/tex]，均有[tex=6.5x1.286]eeb0mFps8SKnQkWxOqAtAKW7+kYhtBAqF1BawLp1vDs=[/tex]（2）[tex=4.357x1.286]Sj/JOIq8D/l1bCecGAbz+Q==[/tex]均为n(n>1)阶方阵，若[tex=6.071x3.643]1eZEffPkvcqd83uGfmH6ioXTGp7TfX9Y4cEwvB0Es2YFcy03+rrxqBOM9P1YKqSU[/tex]，则[tex=9.286x1.286]5GdstVAK2mQUVvrXhqEBXY2ZGYJWvTOSCeQpgWxlaIU=[/tex]（3）[tex=4.357x1.286]Sj/JOIq8D/l1bCecGAbz+Q==[/tex]均为n阶方阵，若[tex=6.071x3.643]1eZEffPkvcqd83uGfmH6ioXTGp7TfX9Y4cEwvB0Es2YFcy03+rrxqBOM9P1YKqSU[/tex]，则[tex=6.714x3.643]oP+BLGjds3y+VSstZfflkb9xRNC7gx/Ut8HlyumVgQ3wpm8lKVEPnHh+4IQyw6I0[/tex]（4）[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]为n(n>1)阶方阵则[tex=9.143x3.643]yKJGwDiIqTdEIZzZw4fSc8b0wn+Z85oKP+XBQo8jFhMJJiT/gspC0k7R1ZucniCQaK9zQzWKkQoOs+eZQ9Pkew==[/tex]（5）[tex=2.0x1.286]cdFQTIcX/k6W15SnnVIOSQ==[/tex]为可逆矩阵，则[tex=4.5x1.286]vwxT16I4QZZNLIV0+aktXg==[/tex]有唯一解[tex=6.5x1.286]2cgMZ4runK/xBXek6dPfdr8+pK73Hbsh9VFZEYYjxE4=[/tex]（6）[tex=9.857x4.643]De166nmeTkb4C/83+ZZH22fNNwRVswy6uT3vQQd8pgT/95tf/antj9R0FEWf59PmQAi09E1UKVl0JJWxdUKuPUyBB+NPd52QqXEQhU15dCD3CSNCTBp8+3ECSZeBeJXaPcbkUgVkPr0ETWsG/kJWpeXrhxVoE+J7OXip6m7dqzSU/N1btrmRg0mGPIxEDbqI2QPJeIh4NWlWy6dfYtyIQA==[/tex]等价于[tex=9.5x4.786]dEdrC9SQsN/3Vx39SaFo4B8usezbIl6DKXEaWVywG/LIYCszsJlNVQJcvKDn5F1NLkIGhJERp247l3yiN0cRXNQA3kiyO/E+P1HrO8u6i0DcWAfv2Pda8l9dQvH/BvFZaWHh73ucJyB581Uia5BEHSePHRdvii+7yDrRZ8nRrBI98gYkXotPisz/zG20foQ9hSRnF6F7YKB7xVURe38tOw==[/tex]
- 若n阶矩阵A存在正整数k，使得[tex=3.0x1.286]k8QEvRxNknQhSHvmEXi5iQ==[/tex]，就称A为幂零矩阵。设幂零矩阵A满足[tex=3.0x1.286]k8QEvRxNknQhSHvmEXi5iQ==[/tex]（k为正整数），试证明I-A可逆，并求其逆矩阵。