Question

如何解下列方程组 要详细过程

Answer 1

增广矩阵 (A, b) =
[1 5 -1 -1 -1]
[1 -2 1 3 3]
[3 8 -1 1 1]
[1 -9 3 7 7]
初等行变换为
[1 5 -1 -1 -1]
[0 -7 2 4 4]
[0 -7 2 4 4]
[0 -14 4 8 8]
初等行变换为
[1 5 -1 -1 -1]
[0 -7 2 4 4]
[0 0 0 0 0]
[0 0 0 0 0]
R(A,, b) = r(A) = 2 < 4, 方程组有无穷多解。
方程组化为
x1 - x3 = -1 - 5x2 + x4
2x3 = 4 + 7x2 - 4x4
取 x2 = x4 = 0, 得特解 (1, 0, 2, 0)^T
导出组为
x1 - x3 = - 5x2 + x4
2x3 = 7x2 - 4x4
取 x2 =2， x4 = 0, 得基础解系 (-3, 2, 7, 0)^T
取 x2 =0， x4 = -1, 得基础解系 (1, 0, 2, -1)^T
方程组的全部接是
x = (1, 0, 2, 0)^T + k (-3, 2, 7, 0)^T + c (1, 0, 2, -1)^T