线性代数第二章答案.docx

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1、第二章矩阵及其运算1.已知线性变换:x12y12y2y3x23y1y25y3x33y12y23y3求从变量x1x2x3到变量y1y2y3的线性变换.解由已知:x221y11y2x2315x3323y2y12211749y1xy23151637y2故x2y2323x3324y3y17x14x29x3y26x13x27x3y33x12x24x32.已知两个线性变换x12y1y3y13z1z2x22y13y22y3y22z1z3x34y1y25y3y3z23z3求从z1z2z到x1xx3的线性变换.32解由已知x

2、1201y1201310x2232y2232201x3415y4150132613z11249z210116z3z1z2z3x16z1z23z3所以有x12z4z9z2123x310z1z216z31111233.设A111,B12432TBA及A111051求AB3AB2A111123111解311112421111110511110581112132230562111217202901114292111123058ATB1111240561110512904.计算下列乘积:4317(1)1232;57

3、01431747321135解123217(2)23165701577201493(2)(123)2;13解(123)2(132231)(10)12(12);(3)1322(1)2224解1(12)1(1)121233(1)3236131(4)2140012;1134131402131解214001267811341312056402a11a12a13x1(5)(x1x2x3)a12a22a23x2;a13a23a33x3解a11a12a13x1(x1x2x3)a12a22a23x2a13a23a33x3a

4、xaxaxaxaxaxaxaxaxx1(12322333)x21112131212213123233x3a11x12a22x22a33x322a12x1x22a13x1x32a23x2x35.设A12,B10问:1312(1)ABBA吗?解ABBA因为AB34BA124638(2)(AB)2A22ABB2吗?解(AB)2A22ABB2所以ABBA因为AB2225(AB)2222281425251429但A22ABB23868101016411812341527所以(AB)2A22ABB2(3)(AB)(A

5、B)A2B2吗?解(AB)(AB)A2B2因为AB22AB022501(AB)(AB)220206250109而A2B23810284113417故(AB)(AB)A2B26.举反列说明下列命题是错误的:（也可参考书上的答案）(1)若A20则A0;解取A0120但A000则A(2)若2,则A0或A;AAE解取A112A,但A0且AE00则A(3)若AX,且A0,则XY.AY解取A10X11Y11001101则AXAY,且A0,但XY.7.设A10,23k1求AAA解A21010101121A3A2A1010

6、1021131Ak10k1108.设A01k,求A.00解首先观察A21010221010102200000023323A3A2A0332003A4A3A443620443004A5A4554103A0554005kkk1k(k1)k2k2A0kkk100k用数学归纳法证明:当k2时,显然成立.假设k时成立，则k1时，kkk1k(k1)k210Ak1AkA02kkk10100k00k1(k1)k1(k1)kk10k1(k2k11)00k1由数学归纳法原理知:kkk1k(k1)k2Ak02kkk1（也可提取公

7、因式，变成书上的答案）00k9.设AB为n阶矩阵，且A为对称矩阵，证明T也是对称矩阵.BAB证明因为ATA所以(T)TT(T)TTTTBABBBABABBAB从而BTAB是对称矩阵.10.设AB都是n阶对称矩阵，证明AB是对称矩阵的充分必要条件是ABBA证明充分性:因为ATABTB且ABBA所以(AB)T(BA)TATBTAB即AB是对称矩阵.必要性:因为ATABTB且(AB)TAB所以AB(AB)TBTATBA11.求下列矩阵的逆矩阵:(1)12;25解A12.

8、A

9、=

10、1,故A-1存在.因为25A*A11A2152,A12A2221故A11A*52.

11、A

12、21(2)cossin;sincos解Acossin-1存在.因为.

13、A故Asincos

14、=110,A*A11A21cossin,A12A22sincos所以A11A*cossin.

15、A

16、sincos121(3)342;541121A342.

17、-1存在.解A

18、=210,故A因为541A*A11A21A31420AAA1361,