西科大网络教育线性代数指导书练习题参考答案.doc

文档介绍：西科大网络教育《线性代数》指导书练习题参考答案
1、计算排列3,2,1,4,5和3,4,1,2,5的逆序数,并说明奇偶性。
答:3>2,3>1,2>1,所以3,2,1,4,5逆序数为3,是奇数;
同理,3>1,3>2,4>1,4>2,所以3,4,1,2,5逆序数为4 ,是偶数。
2、由行列式性质2(P26)知 a 11 a12 a13 a11 a12 a13
10a21 10a22 10a23 =10 a21 a22 a23 =10×2=20
a 31 a32 a33 a31 a32 a33
3、答: 1 -2 5 0 1 -2 5 0 1 -2 5 0 1 -2 5 0
D= -2 3 -8 -1 = 0 -1 2 -1 = 0 -1 2 -1 = 0 -1 2 -1
3 1 -2 4 0 7 -17 4 0 0 -3 -3 0 0 -3 -3
1 4 2 -5 0 6 -3 -5 0 0 9 -11 0 0 0 -20
=1×(-1)×(-3)×(-20)=60
(用行列式性质化上三角行列式)
4、答: 1 0 -1 2 0 1 0 1 2
D= 1 2 0 ,M11= 3 2 =4,M12= -1 2 =2,M13= -1 3 =5
-1 3 2 1
A11=(-1)1+1M11=4,A12=(-1)1+2M12=-2,A13=(-1)1+3M13=5
1 1 1 1 1 4 16 64
5、答:D4= 4 3 7 -5 1 3 9 27
16 9 49 25 = 1 7 49 343 =(-5-4)(-5-3)(-5-7)(7-4)
64 27 343 -125 1 -5 25 -125 (7-3)(3-4)=10368
P42
6、答: 1 2 -1 2 1 2 -1 2 1 2 -1 -8 1 2 -8
D= 3 0 1 5 = 3 0 1 5 =3 0 1 15 =(-1)4+3(-1) 3 0 15
-2 0 3 0 -4 1 1 0 -4 1 11 0–4 11
-2 -4 1 6 0 0 –1 10 0 0 -1 0

1 2 -13 2 -13 2 -13
= 3 0 0 =3×(-1)2+1 =-3 =3×2×(-15)=90
0 -4 11 -4 11 0 -15
(尽可能出现较多0,注意行列变换时,要在前自加“-”号)
7、答: 0 1 1 1 3 1 1 1 1 1 1 1 1 0 0 0
D= 1 0 1 1 = 3 0 1 1 =3 1 0 1 1 =3 1 -1 0 0
1 1 0 1 3 1 0 1 1 1 0 1 1 0 -1 0
1 1 1 0 3 1 1 0 1 1 1 0 1 0 0 -1
=3×1×(-1)×(-1)×(-1)=-3
8、答:x+y-2z=-4 1 1 -2 1 0 0 -7 –31 -7 -31
5x-2y-7z=-7 A= 5 -2 -7 = 5 -7 –31 = = =14
2x-5y-3z=1 2 -5 -3 2 -7 -13 -7 –13 0 -2
-4 1 -2 0 -19 -14 -19 -14 19 14 19 14
A1 = -7 -2 -7 = 0 -37 -28 = = = =14
1 -5 -3 1 -5 -3 -37 -28 37 28 -1 0
1 -4 -2 1 -2 -2 1 -2 -2 5 -7 5 -7
A2 = 5 -7 -7 = 5 0 -7 = 5 0 -7 =(-2)(-1)1+2 =2 =-14
2 1 -3 2 4 -3 4 0 -7 4 -7 -1 0
1 1 -4 1 0 -4 1 0 -4 1 -4 1 -4
A3 = 5 -2 -7 = 5 -7 -7 = 5 -7 -7 =(-7)(-1)2+2 =-7 =28
2 -5 1 2 -7 1 -3 0 8 -3 8 -3 0

由克莱姆法则x = =1, y = =-1, z = =2
x=1
∴线性方程组解为 y=-1
z=2

9、答:设f(x)=ax3+bx2+cx+d (a≠0),由f(0)=0,f(1)=-1,f(2)=4,f(-1)=1
0+0+0+d=0 d=0
得: a+b+c+d=-1 a+b+c=-1 ①
8a+4b+2c+d=4 ∴ 8a+4b+2c=4 ②①+③得2b=0∴b=0
-a+b-c+d=1 -a+b-c=1 ③
a+c=-1 a=1
∴ 8a+2c=4 ∴ c=-2 ∴f(x)=x3-2x
10、答: 1 a1 a12…a1n-1