有一道題

證明線性方程組 X1-X2=a1 X2-X3=a2 X3-X4=a3 x4-x5=a4 X5-X1=a5 有解的充分必要條件是a1+a2+a3+a4+a5=0,

解: 增廣矩陣 =

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

所以方程組有解a1+a2+a3+a4+a5=0

此時, 增廣矩陣 -->

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

方程組的一般解為:

(a1+a2+a3+a4, a2+a3+a4,a3+a4, a4, 0)" + c(1,1,1,1,1)".