Question

用高斯消元法求解方程组的解：2A-B-C+D=2 ,A+B-2C+D=4 ,4A-6B+2C-2D=4 ,3A+6B-9

Answer 1

增广矩阵 =

2 -1 -1 1 2

1 1 -2 1 4

4 -6 2 -2 4

3 6 -9 7 9

r4-r1-r2,r3-2r1,r1-2r2

0 -3 3 -1 -6

1 1 -2 1 4

0 -4 4 -4 0

0 6 -6 5 3

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

0 0 0 2 -6

1 0 -1 0 4

0 1 -1 1 0

0 0 0 3 -9

r1*(1/2),r3-r1,r4-3r1

0 0 0 1 -3

1 0 -1 0 4

0 1 -1 0 3

0 0 0 0 0

交换行

1 0 -1 0 4

0 1 -1 0 3

0 0 0 1 -3

0 0 0 0 0

方程组的通解为 (4,3,-3,0)^T+k(1,1,1,0)^T.