Question

设线性方程组为X1-X2=A1 求证方程组有解得充分必要条件是5seigema(i=1)=0

Answer 1

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

所有行加到第5行

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 ∑Ai

因为方程组有解的充分必要条件是系数矩阵的秩等于增广矩阵的秩

所以 此方程组有解的充分必要条件是 ∑Ai = 0.