吉林大学线性代数线性习题2.ppt

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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()()ABABBAABBAAB

2、AABBAB 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()110521631616(

3、)()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1254613

4、21254613213546122122308XX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(2)(4,1,4)(,1,)441 118(,)

5、(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111112222288422112341234|()AEEBAA BBOAOBOA BAAAACCOACCBOCCOAEOCCBOOE