已知线性方程组(x1+x2+2X3+3X4=1;X1+3X2+6X3+X4=3;X1-5X2-10X3+9X4=a)

Question

问a为何值时方程组有解？2. 当方程组有解时求出它的全部解

Answer 1

x1+x2+2X3+3X4=1
X1+3X2+6X3+X4=3
X1-5X2-10X3+9X4=a

增广矩阵=
1 1 2 3 1
1 3 6 1 3
1 -5 -10 9 a

r2-r1,r3-r1
1 1 2 3 1
0 2 4 -2 2
0 -6 -12 6 a-1

r3+3r2,r2*(1/2),r1-r2
1 0 0 4 0
0 1 2 -1 1
0 0 0 0 a+5

所以 a=-5 时方程组有解
通解为 (0,1,0,0)^T+k1(0,-2,1,0)^T+k2(-4,1,0,1)^T