解下列线性方程组,在有无穷多解的情况下,求其通解2x1-x2+4x3-3x4=-4x1+x3-x4=-33x1+x2+x3=17x1+7x3-3x4=3

Question

解下列线性方程组,在有无穷多解的情况下,求其通解
2x1-x2+4x3-3x4=-4
x1+ x3-x4=-3
3x1+x2+x3 =1
7x1+ 7x3-3x4=3

Answer 1

增广矩阵 =
2 -1 4 -3 -4
1 0 1 -1 -3
3 1 1 0 1
7 0 7 -3 3
r1-2r2,r3-3r2,r4-7r2
0 -1 2 -1 2
1 0 1 -1 -3
0 1 -2 3 10
0 0 0 4 24
r1+r3
0 0 0 2 12
1 0 1 -1 -3
0 1 -2 3 10
0 0 0 4 24
r1*(1/2),r2+r1,r3-3r1,r4-r1
0 0 0 1 6
1 0 1 0 3
0 1 -2 0 -8
0 0 0 0 0
交换行
1 0 1 0 3
0 1 -2 0 -8
0 0 0 1 6
0 0 0 0 0
通解为:(3,-8,0,6)^T+c(-1,2,1,0)^T.