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

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