线性代数问题（关于方程组有解的条件）设有方程组x1-x2=a1,x2-x3=a2,x3-x4=a3,x4-x5=a4,x5-x1=a5.证明：方程组有解的充要条件是a1+a2+a3+a4+a5=0
网友回答
方程组有解的充分必要条件是系数矩阵与增广矩阵有相同的秩
系数矩阵为1 -1 0 0 0 1 -1 0 0 0
0 1 -1 0 0 0 1 -1 0 0
0 0 1 -1 0 = 0 0 1 -1 0 系数矩阵的秩为4
0 0 0 1 -1 0 0 0 1 -1
-1 0 0 0 1 0 0 0 0 0
增广矩阵为1 -1 0 0 0 a1 1 -1 0 0 0 a1
0 1 -1 0 0 a2 0 1 -1 0 0 a2
0 0 1 -1 0 a3 = 0 0 1 -1 0 a3
0 0 0 1 -1 a4 0 0 0 1 -1 a4
-1 0 0 0 1 a5 0 0 0 0 0 a1+a2+a3+a4+a5
满足增广矩阵的秩为4,则a1+a2+a3+a4+a5=0