求下列方程组何时有解,并在有解时求其全部解：X1+2X2-ax3=-1x1+2x2+2x3=b4x1+5x2+10x3=-1

Question

求下列方程组何时有解,并在有解时求其全部解：
X1+2X2-ax3=-1
x1+2x2+2x3=b
4x1+5x2+10x3=-1

Answer 1

增广矩阵 (A,b) =
1 2 -a -1
1 2 2 b
4 5 10 -1
r1-r2,r3-4r2
0 0 -a-2 -1-b
1 2 2 b
0 -3 2 -1-4b
交换行
1 2 2 b
0 -3 2 -1-4b
0 0 -a-2 -1-b
当a≠-2时有唯一解.
当a=-2,b≠-1时无解
当a=-2,b=-1时有无穷多解