答案
?100???11?1一、（1）10;(2)(A?2I)???0?;(3)A?0;?22?001??1??10??2??131|A|22(4)??2??,f?x1?x3?2x1x2?x1x3?4x2x3;(5),??22????0?320???二、（1）C；（2）A；（3）D；（4）A；（5）B?8三、1.由AB?A?B?(A?I)B?A?B?(A?I)?1A???1?21???81?100?1??A?B???472?????627???2?1011??10?112?2.(A|b)???3?2110?????01?213????10?11?????0000??2????1???1??2????2时,方程组有解,且通解为:x?k?2?????1????3??1??1??k2??????0?00???1?????0????????1?22?3.(1)二次型f的矩阵A????2?24????24?2??(2)|A??I|??(7??)(2??)2,?A的特征值为2,2,?7.对应??2的线性无关的特征向量是??????2??1????2??1??,?2??0??0????1??
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?2??1??4????2??2???????????(?1,?2)?1??把?1，?2正交化，?1??1??1?，?2??2????1??4?(?1,?1)5???0????5???2??2????????1?11?1??再把?1,?2单位化：p1???1??1?,p2???2??4?||?1||||?||5??35??20???5??1?1??对应于??7的单位特征向量为p3??2?3????2?令P?（p1p2p3），则正交变换为x?Py222f的标准形为f?x?Ax?y?P?APy?2y1?2y2?7y3*?1?1?1?1?1
四、1.证:(AB)?|AB|(AB)?|A|?|B|?B?A?|B|?B?|A|?A?B*?A*???????22kk2.证:A???0?,A???0A???0?,......,A???0?
五、A不能与一个对角矩阵相似.反证:若存在可逆矩阵P使P?1AP???A?P?P?1,Ak?P?kP?1,但Ak?0?P?kP?1?0??k?0??的对角元素全为零???0?A?0,这与A是非零矩阵矛盾.