第二章 线性方程组习题答案与解答

第二章 线性方程组习题答案与解答

习题二

对于数字计算题，仅给出Maple 程序与答案.证明题答案仅供参考。 1.用消元法解下列方程组

(1)1221231231321,22,353,22

x x x x x x x x x x x -+=⎧⎪

--=⎪⎨-+=⎪⎪-++=-⎩ > A:=[[1,-1,2],[1,-2,-1],[3,-1,5],[-1,0,2]]: b:=[1,2,3,-2]:linsolve(A,b);

⎡⎣⎢⎢⎤⎦⎥⎥

,,107-17-27

12345122451

23452322,

(2)3536,2228.

x x x x x x x x x x x x x x x -+-+=⎧⎪

-+-+=⎨⎪++--=⎩

> A:=[[1,-2,3,-1,2],[3,-1,5,-3,1],[2,1,2,-2,-1]]: b:=[2,6,8]: linsolve(A,b);

(3) >

A:=[[1,2,3],[3,5,7],[2,3,4]]:b:=[4,9,5]:linsolve(A,b,'r',c);

[],,- + 2c 1 - 32c 1c 1

(4)

> A:=[[2,-2,1,-1,1],[1,-4,2,-2,3],[3,-6,1,-3,4],[1,2,-1,1,-2]]:

b:=[2,3,5,-1]: linsolve(A,b,'r',c);

⎡⎣⎢⎢⎤⎦

⎥⎥,,,, + 1313c 1c 20- - + 432c 253c 1c 1 (5) > A:=[[1,1,2,3],[2,3,5,2],[3,-1,-1,-2],[3,5,2,-2]]:

b:=[1,-3,-4,-10]: linsolve(A,b,'r',c);

[],,,-1-101

(6)

> A:=[[2,-4,5,3],[3,-6,4,2],[4,-8,17,11]]: b:=[0,0,0]:

linsolve(A,b,'r',c);

⎡⎣⎢⎢⎤⎦

⎥⎥,,, - 2c 125c 2c 1c 2-75c 2

(7)

> A:=[[1,3,-2,-1],[2,6,-3,0],[3,9,-9,-5]]: b:=[3,13,8]:

linsolve(A,b,'r',c);

[],,, - 23c 1c 1-35

(8)

> A:=[[1,-1,2,-3,1],[2,-2,7,-10,5],[3,-3,3,-5,0]]: b:=[2,5,5]:

linsolve(A,b,'r',c);

⎡⎣⎢⎢⎤⎦

⎥⎥,,,, + - + 53c 2c 353c 1c 2c 3c 1- + + c 343c 113

2.当k 取何值时,下面的齐次线性方程组有非零解,并求出此非零解.

> A:=matrix([[2,-1,3],[3,-4,7],[-1,2,k]]); E:=matrix([[1,0,0],[0,1,0],[0,0,1]]); k1:=solve(det(A)=0,k);

A:=matrix([[2,-1,3],[3,-4,7],[-1,2,-3]]); b:=[0,0,0]:

linsolve(A,b,'r',c);

:= A ⎡⎣⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥2-133-47-12k := E ⎡⎣⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥100010001 := k1-3 := A ⎡⎣⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥2-133-47-12-3 []

,,-c 1c 1c 1

3. 当k 取何值时,下面的线性方程组无解?有解?,在方程组有解时,求出它

的解.

.4.当a 取何值时,线性方程组

1231231

231,233,32

x x x x x ax x ax x +-=⎧⎪

++=⎨⎪++=⎩ 无解?有唯一解?有无穷多解?在方程组有解时,求出它的解. 11111

1112330121.1320141a a a a --⎛⎫⎛⎫ ⎪ ⎪→+ ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭

1.1⎫⎪⎭

123210,1,

1111111

10131013100410011/41105/410010101/40101/4.0011/40011/41,1/4,1/4.10,1,

111111110121012101410062a a x x x a a a a a a a a -==--⎛⎫⎛⎫ ⎪ ⎪

→ ⎪ ⎪ ⎪ ⎪⎝⎭⎝

⎭⎛⎫⎛⎫

⎪ ⎪

→→ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭

===-≠≠--⎛⎫⎛ ⎪ +→+ ⎪ ⎪---+-⎝⎭⎝11

110121.00(2)(3)2(2)(3)0,23,11

11111

10121012100(2)(3)20031111111

04/(3)01210101/(3)0011/(3)0011/(3)a a a a a a a a a a a a a a a a a a a ⎫⎪⎪

⎪⎭

-⎛⎫ ⎪=+ ⎪ ⎪-+-⎝⎭

-+≠≠≠---⎛⎫⎛⎫

⎪ ⎪+→+ ⎪

⎪ ⎪ ⎪-+-+⎝⎭⎝

⎭-+⎛⎫⎛ ⎪ →+→+ ⎪ ⎪ ++⎝⎭⎝即且时123

1003/(3)0101/(3),0011/(3)3/(3),1/(3),1/(3).

a a a x a x a x a ⎫

⎪

⎪⎪⎭

+⎛⎫ ⎪→+ ⎪ ⎪+⎝⎭

=+=+=+唯一解