求方程组的通解2X1-3X2+X3-5X4=1,-5X1-10X2-2X3+X4=-21,X1+4X2+3X3+2X4=1,2X1-4X2+9X3-3X4=-16.麻烦啊

Question

求方程组的通解
2X1-3X2+X3-5X4=1,-5X1-10X2-2X3+X4=-21,X1+4X2+3X3+2X4=1,2X1-4X2+9X3-3X4=-16.
麻烦啊

Answer 1

增广矩阵 =
2 -3 1 -5 1
-5 -10 -2 1 -21
1 4 3 2 1
2 -4 9 -3 -16
r1-r4,r2+5r3,r4-2r3
0 1 -8 -2 17
0 10 13 11 -16
1 4 3 2 1
0 -12 3 -7 -18
r2-10r1,r3-4r1,r4+12r1
0 1 -8 -2 17
0 0 93 31 -186
1 0 35 10 -67
0 0 93 31 -186
r4-r2,r2*(1/93)
0 1 -8 -2 17
0 0 1 1/3 -2
1 0 35 10 -67
0 0 0 0 0
r1+8r2,r3-35r2
0 1 0 2/3 1
0 0 1 1/3 -2
1 0 0 -5/3 3
0 0 0 0 0
交换行
1 0 0 -5/3 3
0 1 0 2/3 1
0 0 1 1/3 -2
0 0 0 0 0
方程组的全部解为:(3,1,-2,0)^T + c(5,-2,-1,3)^T.