Question

写出方程组x1-x2=a1,x2-x3=a2,x3-x4=a3,x4-x5=a4,x5-x1=a5有解的充要条件，并求其解

Answer 1

有一道题
证明线性方程组 X1-X2=a1 X2-X3=a2 X3-X4=a3 x4-x5=a4 X5-X1=a5 有解的充分必要条件是a1+a2+a3+a4+a5=0,
解: 增广矩阵 =
1 -1 0 0 0 a1
0 1 -1 0 0 a2
0 0 1 -1 0 a3
0 0 0 1 -1 a4
-1 0 0 0 1 a5

r5+r1+r2+r3+r4
1 -1 0 0 0 a1
0 1 -1 0 0 a2
0 0 1 -1 0 a3
0 0 0 1 -1 a4
0 0 0 0 0 a1+a2+a3+a4+a5

所以方程组有解<=>a1+a2+a3+a4+a5=0
此时, 增广矩阵 -->
1 -1 0 0 0 a1
0 1 -1 0 0 a2
0 0 1 -1 0 a3
0 0 0 1 -1 a4
0 0 0 0 0 0

r3+r4, r2+r3,r1+r2
1 0 0 0 -1 a1+a2+a3+a4
0 1 0 0 -1 a2+a3+a4
0 0 1 0 -1 a3+a4
0 0 0 1 -1 a4
0 0 0 0 0 0

方程组的一般解为:
(a1+a2+a3+a4, a2+a3+a4,a3+a4, a4, 0)' + c(1,1,1,1,1)'.

Answer 2

a1+a2+a3+a4+a5=0

解的无数组