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

方程组的一般解是指所有解,
又称通解.
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
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<=>,1,1,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,1)':
(a1+a2+a3+a4,
a2+a3+a4,a3+a4,
a4,
0)'
+
c(1,1