Question

用初等行变换来解下列线性方程组

（1）2x1-x2+3x3=3
3x1+x2-5x3=0
4x1-x2+x3=3
x1+3x2-13x3=-6

(2) x1-2x2+x3+x4=1
x1-2x2+x3-x4=-1
x1-2x2+x3-5x4=5

(3) x1-x2+x3-x4=1
x1-x2-x3+x4=0
x1-x2-2x3+2x4=-1/2

Answer 1

(1) 解: 增广矩阵 =
2 -1 3 3
3 1 -5 0
4 -1 1 3
1 3 -13 -6

r3-2r1,r2-r1-r4,r1-2r4
0 -7 29 15
0 -1 5 3
0 1 -5 -3
1 3 -13 -6

r1+7r3,r2+r3,r4-3r3
0 0 -6 -6
0 0 0 0
0 1 -5 -3
1 0 2 3

r1*(-1/6),r3+5r1,r4-2r1
0 0 1 1
0 0 0 0
0 1 0 2
1 0 0 1

交换行
1 0 0 1
0 1 0 2
0 0 1 1
0 0 0 0

方程组的解为: x1=1,x2=2,x3=1.