Question

a为何值时时,线性方程组{2x1+5x2+5x3+4x4=10 x1+2x2+2x3+x4=3 x1+3x2+3x3+3

Answer 1

增广矩阵(A,b)=

2 5 5 4 10

1 2 2 1 3

1 3 3 3 a

r1-2r2,r3-r2

0 1 1 2 4

1 2 2 1 3

0 1 1 2 a-3

r2-2r1,r3-r1

0 1 1 2 4

1 0 0 -3 -5

0 0 0 0 a-7

r1r2

1 0 0 -3 -5

0 1 1 2 4

0 0 0 0 a-7

当a=7时方程组有解且有无穷多解

此时方程组的通解为:(-5,4,0,0)^T+c1(0,-1,1,0)^T+c2(3,-2,0,1)^T