求下列线性方程组的通解:2x1+x2-x3+x4=1,4x1+2x2-2x3+x4=2,2x1+x2-x3-x4=1

问题描述:

求下列线性方程组的通解:2x1+x2-x3+x4=1,4x1+2x2-2x3+x4=2,2x1+x2-x3-x4=1

增广矩阵 =
2 1 -1 1 1
4 2 -2 1 2
2 1 -1 -1 1
r2-2r1,r3-r1
2 1 -1 1 1
0 0 0 -1 0
0 0 0 -2 0
r1+r2,r3-2r2,r2*(-1),的*(1/2)
1 1/2 -1/2 0 1/2
0 0 0 1 0
0 0 0 0 0
通解为:(1/2,0,0,0)+c1(-1/2,1,0,0)+c2(1/2,0,1,0),c1,c2 为任意常数