吉林大学线性代数线性习题2

1、习题2矩阵计算乘积43173512326570149 3(1,2,3) 2101 2241 ( 1,2)12336 计算乘积n二次型11121311232122232313233322211 1122112133113222233223333(,)()()()aaaxx xxaaaxaaaxa xaax xaax xa xaax xa x3、线性变换复合113212331231122133332232453232013106132322011249415013101 16xyyxyyyxyyyyzzyzzyzz 交换222221210,13123412,4636()2()()ababbaabb

2、aabaabbab abab反例20010aoaoa反例2,11221122aa ao aea 反例,1 11021,1 10110axay ao xyaxy高次幂计算212222110,?1100+110100110101kkkkkkkkaaaebboaekebc eekebek bbkk可交换高次幂计算2312223312221231111111111 ,()()()()()11111nnnnnnnnnnnnnnnnaebbbboaebenebcebceenbcbbnc212211nnnnnnnncn可交换补充一种情况223211110521052010 ,?33015()11052163

3、1616()()16105161620103015nnnnaaaaaaaaaa tttttttttttttabab aba b a bb aabab ab aba b a b a b证明对称矩阵()()tttttttttaab abb abb abb ab对称证明对称矩阵()tttttaabbabbaabb abaabab对称求逆矩阵*1*1225| 1522152121|aaaaaa求逆矩阵11cossinsincoscos()sin()sin()cos()cossinsincos()taa正定矩阵旋转的逆变换=顺时针旋转变换求逆矩阵n对角矩阵的逆11111221nnaaaaaa解矩阵方程1

4、25461321254613213546122122308xx解矩阵方程11111431120111120114311201111201114311201111201201114311201210134xx102求行列式*111*111311|21|215| (2 )5| | | 2| ( 2) |16221112aaa aaaaaaaaa 验证：此题书后答案有误矩阵方程1222(2 )(2 )()()()ababae babaeaabeabae baeaeaebae矩阵方程*111*1*1128(2 )88(2 )(1, 2,1)1(1,1)21|( 2)(1,1)( 2,1, 2)211(

5、2 )( 4, 1, 4)(, 1,)441 118(,)(2, 4,2)4 24a babaeae baebaeaadiagadiagaa adiagdiagaediagdiagbdiagdiag 伴随矩阵推导原矩阵*131*1111| |8 |11 1 1( ,4)|2 2 21(2,2,2, )43()33()naaaaadiagaadiagababaeae baebaea证明题121212121()()() kkkkkkkaoeaeaaaeeaa eaaaaaaaaaeea抽象 逆矩阵21212()21()2()()641(2 )2(3 )43aaeoa aeeaaeaeaeaaeeaeae 伴随和逆*11 *1*1111 *1111*11 *()()|1()(|)|()| ()|()()aaaa aaa aaaaaaaaaa伴随矩阵的行列式*1*1*1*1*1111):,2 :,| 0| 0|,| 0| 0| 0 | 0| det(|) | | | 0| 0|nnnnaoaoaaaaa eoaaoaaaaaaaa aaa aaaaaaaa*结论显然） 假设 可逆，则从右侧乘以(a )可以得到，矛盾！所以有不可逆，所以其他1111122222