当A为何值时,线性方程组有唯一解,无穷解,无解,要求详解联立一个方程组:(A+1)*X1+2*X2-X3=-13*X1+(A+1)*X2-2*X3=1-3*X1+4*X2+(A+1)*X3=1

Question

当A为何值时,线性方程组有唯一解,无穷解,无解,要求详解
联立一个方程组:
(A+1)*X1+2*X2-X3=-1
3*X1+(A+1)*X2-2*X3=1
-3*X1+4*X2+(A+1)*X3=1

Answer 1

系数行列式|A| =
λ+1 2 -1
3 λ+1 -2
-3 4 λ+1
=λ(λ+1)(λ+2).
所以当 λ≠0 且 λ≠-1 且 λ≠-2 时方程组有唯一解.
当λ=0时,增广矩阵 =
1 2 -1 -1
3 1 -2 1
-3 4 1 1
r2-r1,r3+r1
1 2 -1 -1
1 -3 0 3
-2 6 0 0
r2+(1/2)r3
1 2 -1 -1
0 0 0 3
-2 6 0 0
此时方程组无解
当λ=-1时,增广矩阵 =
0 2 -1 -1
3 0 -2 1
-3 4 0 1
r3-2r1+r2
0 2 -1 -1
3 0 -2 1
0 0 0 4
此时方程组无解
当λ=-2时,增广矩阵 =
-1 2 -1 -1
3 -1 -2 1
-3 4 -1 1
r2-r1,r3-r1
-1 2 -1 -1
5 -5 0 3
-2 2 0 2
r2+(5/2)r3
-1 2 -1 -1
0 0 0 8
-2 2 0 2
此时方程组无解
好麻烦这个