Question

求非齐次线性方程组的通解

x1+3x2+5x3-4x4=1
x1+3x2+2x3-2x4+x5=-1
x1-2x2+x3-x4-x5=3
x1+2x2+x3-x4-x5=3

Answer 1

解: 增广矩阵 =
1 3 5 -4 0 1
1 3 2 -2 1 -1
1 -2 1 -1 -1 3
1 2 1 -1 -1 3

r4-r3, r4*(1/4), r1-3r4,r2-3r4,r3+2r4
1 0 5 -4 0 1
1 0 2 -2 1 -1
1 0 1 -1 -1 3
0 1 0 0 0 0

r1-r3,r2-r3
0 0 4 -3 1 -2
0 0 1 -1 2 -4
1 0 1 -1 -1 3
0 1 0 0 0 0

r1-4r2,r3-r2
0 0 0 1 -7 14
0 0 1 -1 2 -4
1 0 0 0 -3 7
0 1 0 0 0 0

r2+r1
0 0 0 1 -7 14
0 0 1 0 -5 10
1 0 0 0 -3 7
0 1 0 0 0 0

交换行
1 0 0 0 -3 7
0 1 0 0 0 0
0 0 1 0 -5 10
0 0 0 1 -7 14

通解为: (7,0,10,14,0)'+c(3,0,5,7,1)'

Answer 2

非齐次线性方程组求通解

Answer 3

k * [7,0,14,10,0]T + [10,0,15,21,1]T