Question

当p,q为何值时,线性方程组x1+x2+x3+x4+x5=0 ,3x1+2x2+x3+x4-3x

=p,x2+2x3+2x4+6x5=3, 5x1+4x2+3x3+3x4-x5=q有解并在有解时求出其全部解

Answer 1

增广矩阵=
1 1 1 1 1 0
3 2 1 1 -3 p
0 1 2 2 6 3
5 4 3 3 -1 q

r3-2r1-r2,r2-3r1
1 1 1 1 1 0
0 -1 -2 -2 -6 p
0 1 2 2 6 3
0 0 0 0 0 q-p

r1-r3,r2+r3
1 0 -1 -1 -5 -3
0 0 0 0 0 p+3
0 1 2 2 6 3
0 0 0 0 0 q-p

所以 p=q=-3时方程组有解
通解为 (-3,3,0,0,0)^T+k1(1,-2,1,0,0)^T+k2(1,-2,0,1,0)^T+k3(5,-6,0,0,1)^T