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1、线性代数(清华版)部分习题参考答案赵燕芬武汉大学数学与统计学院Email:2008.122目录第一章行列式5第二章矩阵21第三章线性方程组41第四章向量空间与线性变换61第五章特征值与特征向量7934目录第一章行列式计算下列数字元素行列式10.flflflflflflflflflflflflflflflfl0001000200 .0800090000000010flflflflflflflflflflflflflflflfl= 10 (1)10+10flflflflflflflflflflflflfl00010020 .08009000flflflflflflflflflflflflfl= 1
2、0 (1)98 29! = 10!.11.flflflflflflflflflfl1111111111111111flflflflflflflflflflrir1= = = = = =i=2,3,4flflflflflflflflflfl1111020000200002flflflflflflflflflfl= 8.12.flflflflflflflflflfl1234234134124123flflflflflflflflflflr1+ri= = = = = =i=2,3,4flflflflflflflflflfl10101010234134124123flflflflflflflflflf
3、l= 10flflflflflflflflflfl1111234134124123flflflflflflflflflflcici1= = = = = =i=4,3,210flflflflflflflflflfl1000211331314311flflflflflflflflflfl按r1= = = = = 展开10flflflflflflflfl113131311flflflflflflflflr1+ri= = = = =i=2,310flflflflflflflfl111131311flflflflflflflflcic3= = = = =i=1,210flflflflflflflfl00
4、1041401flflflflflflflfl= 160.14.flflflflflflflflflflflflfl3656425453363422546511111flflflflflflflflflflflflflr1r5= = = = = flflflflflflflflflflflflfl1111125453363422546536564flflflflflflflflflflflflflr2r12,r3r13= = = = = = = = = = = = =r4r12,r5r13flflflflflflflflflflflflfl1111103275030750328703297fl
5、flflflflflflflflflflflflrir2= = = = = =i=3,4,5flflflflflflflflflflflflfl1111103275002000001200022flflflflflflflflflflflflflr5r42= = = = = = =flflflflflflflflflflflflfl1111103275002000001200002flflflflflflflflflflflflfl= 12.16.flflflflflflflflflflflflfl12345678910000130002401011flflflflflflflflflflfl
6、flflr3r5= = = = = flflflflflflflflflflflflfl12345678910010110002400013flflflflflflflflflflflflfl= flflflflflflflfl123678010flflflflflflflflflflflflfl2413flflflflfl= (1)3+2 (8 18) (6 4) = 20.56第一章 行列式17.flflflflflflflflflflflflfl0011200302002401240131258flflflflflflflflflflflflfl= (1)32flflflflflflfl
7、fl112302240flflflflflflflflflflflflfl1231flflflflfl= 5flflflflflflflfl112210240flflflflflflflfl= 60.证明下列恒等式19.由行列式的线性性质， 可将左边的行列式拆分为4个行列式的和,即flflflflflflflfla1+ b1xa1x + b1c1a2+ b2xa2x + b2c2a3+ b3xa3x + b3c3flflflflflflflfl=flflflflflflflfla1a1xc1a2a2xc2a3a3xc3flflflflflflflfl+flflflflflflflfla1b1c
8、1a2b2c2a3b3c3flflflflflflflfl+flflflflflflflflb1xa1xc1b2xa2xc2b3xa3xc3flflflflflflflfl+flflflflflflflflb1xb1c1b2xb2c2b3xb3c3flflflflflflflfl= 0 +flflflflflflflfla1b1c1a2b2c2a3b3c3flflflflflflflflflflflflflflflfla1xb1xc1a2xb2xc2a3xb3xc3flflflflflflflfl+ 0 = (1 x2)flflflflflflflfla1b1c1a2b2c2a3b3c3flf
9、lflflflflflfl=右边.20.flflflflflflflflflfl1 + x11111 x11111 + y11111 yflflflflflflflflflfl=flflflflflflflflflfl1 + x1 + 01 + 01 + 01 + 01 x1 + 01 + 01 + 01 + 01 + y1 + 01 + 01 + 01 + 01 yflflflflflflflflflfl=flflflflflflflflflfl10001x0010y0100yflflflflflflflflflfl+flflflflflflflflflflx100010001y0010yf
10、lflflflflflflflflfl+flflflflflflflflflflx0100x100010001yflflflflflflflflflfl+flflflflflflflflflflx0010x0100y10001flflflflflflflflflfl+flflflflflflflflflflx0000x0000y0000yflflflflflflflflflfl= xy2 xy2+ x2y x2y + x2y2= x2y2.20.另证： 若x = 0或y = 0 ,等式显然成立.当xy 6= 0时,flflflflflflflflflfl1 + x11111 x11111 +
11、y11111 yflflflflflflflflflfl=flflflflflflflflflflflflfl1111101 + x111011 x110111 + y101111 yflflflflflflflflflflflflflrir1= = = = = = =i=2,3,4,5flflflflflflflflflflflflfl111111x00010x00100y01000yflflflflflflflflflflflflflc1+1 xc2 c1+1 xc3= = = = = = = = = = c1+1 yc4 c1+1 yc5flflflflflflflflflflflflf
12、l111110x00000x00000y00000yflflflflflflflflflflflflfl= x2y2.21.flflflflflflflfl111abca3b3c3flflflflflflflflc2c1= = = = =c3c1flflflflflflflfl100ab ac aa3b3 a3c3 a3flflflflflflflfl= (b a)(c3 a3) (c a)(b3 a3)= (b a)(c a)(c2+ ac + a2 b2 ab a2) = (b a)(c a)(c b)(a + b + c).722.flflflflflflflfl1a2a31b2b31c
13、2c3flflflflflflflflr2r1= = = = =r3r1flflflflflflflfl1a2a30b2 a2c2 a20c2 a2c3 a3flflflflflflflfl= (b2 a2)(c3 a3) (c2 a2)(b3 a3)= (b a)(c a)(b + a)(c2+ ac + a2) (c + a)(b2+ ab + a2) = (b a)(c a)bc2+ ac2 b2c ab2= (b a)(c a)(c b)(ab + bc + ca) = (ab + bc + ca)flflflflflflflfl1aa21bb21cc2flflflflflflflfl
14、.计算下列各题23.flflflflflflflflflfl102a20b03c45d000flflflflflflflflflfl= d (1)4+1flflflflflflflfl02a0b0c45flflflflflflflfl= d c (1)3+1flflflflfl2ab0flflflflfl= dc (0 ab) = abcd.24.flflflflflflflflflfla1001b1001c1001dflflflflflflflflflfl= aflflflflflflflflb101c101dflflflflflflflflflflflflflflflfl1100c101d
15、flflflflflflflfl= a(bcd + b + d) (cd) = abcd + ab + ad + cd.25.flflflflflflflflflfla2(a + 1)2(a + 2)2(a + 3)2b2(b + 1)2(b + 2)2(b + 3)2c2(c + 1)2(c + 2)2(c + 3)2d2(d + 1)2(d + 2)2(d + 3)2flflflflflflflflflflcici1= = = = = =i=4,3,2flflflflflflflflflfla22a + 12a + 32a + 5b22b + 12b + 32b + 5c22c + 12c + 32c + 5d22d + 12d + 32d + 5flflflflflflflflflflcic2= = = = =i=3,4flflflflflflflflflfla22a + 124b22b + 124c22c + 124d22d + 124flflflflflflflflflfl= 0.26.flflflflflflflflflflflabc1bca1cab1 b + c 2
