求下列方程组何时有解,并在有解时求其全部解：要求写成解题步骤x1+2x2+x3=12x1+3x2+(a+2)x3=3x1+ax2-2x3=0

Question

求下列方程组何时有解,并在有解时求其全部解：要求写成解题步骤
x1+2x2+x3=1
2x1+3x2+(a+2)x3=3
x1+ax2-2x3=0

Answer 1

增广矩阵 (A,b) =
1 2 1 1
2 3 a+2 3
1 a -2 0
r2-2r1,r3-r1
1 2 1 1
0 -1 a 1
0 a-2 -3 -1
r3+(a-2)r2
1 2 1 1
0 -1 a 1
0 0 (a-3)(a+1) a-3
所以 a≠3 且 a≠-1 时,方程组有唯一解.
a=-1时,方程组无解
a=3时,方程组有无穷多解.