px1+x2+x3=1, x1+px2+x3=p, x1+x2+px3=p².当p取何值时,方程组有唯一解,无解,有无穷解并求通解.

px1+x2+x3=1, x1+px2+x3=p, x1+x2+px3=p².当p取何值时,方程组有唯一解,无解,有无穷解并求通解.
px1+x2+x3=1, x1+px2+x3=p, x1+x2+px3=p².当p取何值时,方程组有唯一解,无解,有无穷解并求通解.

px1+x2+x3=1, x1+px2+x3=p, x1+x2+px3=p².当p取何值时,方程组有唯一解,无解,有无穷解并求通解.
解: 系数行列式|A| = (λ+2)(λ-1)^2.
所以当 λ≠1 且 λ≠-2 时方程组有唯一解.
当λ=1时, 增广矩阵 =
1 1 1 1
1 1 1 1
1 1 1 1
r2-r1,r3-r1
1 1 1 1
0 0 0 0
0 0 0 0
此时方程组有无穷多解: (1,0,0)'+c1(-1,1,0)'+c2(-1,0,1)'.
当λ=-2时, 增广矩阵 =
-2 1 1 1
1 -2 1 -2
1 1 -2 4
r3+r1+r2
-2 1 1 1
1 -2 1 -2
0 0 0 3
此时方程组无解.