解: 增广矩阵=
1 -2 -1 -1 2
2 -4 5 3 0
3 -6 4 3 3
4 -8 17 11 λ
r2-2r1,r3-3r1,r4-4r1
1 -2 -1 -1 2
0 0 7 5 -4
0 0 7 6 -3
0 0 21 15 λ-8
r3-r2,r4-3r2
1 -2 -1 -1 2
0 0 7 5 -4
0 0 0 1 1
0 0 0 0 λ+4
所以, λ=-4时方程组有解
此时, 继续化增广矩阵为行简化梯矩阵
1 -2 -1 -1 2
0 0 7 5 -4
0 0 0 1 1
0 0 0 0 0
r1+r3,r2-5r3
1 -2 -1 0 3
0 0 7 0 -9
0 0 0 1 1
0 0 0 0 0
r2*(1/7), r1+r2
1 -2 0 0 12/7
0 0 1 0 -9/7
0 0 0 1 1
0 0 0 0 0
方程组的通解为: (12/7,0,-9/7,1)'+c(2,1,0,0)'