a为何值时,有唯一解,无解,无穷多解x1-2x2+3x3=-12x2-x3=2a(a-1)x3=(a-1)(a+2)

Question

a为何值时,有唯一解,无解,无穷多解
x1-2x2+3x3=-1
2x2-x3=2
a(a-1)x3=(a-1)(a+2)

Answer 1

增广矩阵 (A,b) =
1 -2 3 -1
0 2 -1 2
0 0 a(a-1) (a-1)(a+2)
当 a≠0,a≠1 时,r(A) = 3 = r(A,b),所以有唯一解.
当 a=0 时,r(A) = 2 ≠ 3 = r(A,b),方程组无解.
当 a = 1 时,r(A) = 2 = r(A,b) < 3,方程组有无穷多解.