Question

求解下列线性方程组 求详细的解答

Answer 1

增广矩阵 (A, b) =
[1 1 1 1 1 7]
[3 2 1 1 -3 -2]
[0 1 2 0 6 23]
[5 4 -3 3 -1 12]
初等行变换为
[1 1 1 1 1 7]
[0 -1 -2 -2 -6 -23]
[0 1 2 0 6 23]
[0 -1 -8 -2 -6 -23]
初等行变换为
[1 0 -1 -1 -5 -16]
[0 1 2 2 6 23]
[0 0 0 -2 0 0]
[0 0 -6 0 0 0]
初等行变换为
[1 0 0 0 -5 -16]
[0 1 0 0 6 23]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
r(A, b) = r(A) = 4 < 5, 方程组有无穷多解。
方程组化为
x1 = -16 + 5x5
x2 = 23 - 6x5
x3 = 0
x4 = 0
取 x5 = 0, 得特解 (-16, 23, 0, 0, 0)^T;
导出租是
x1 = 5x5
x2 = -6x5
x3 = 0
x4 = 0
取 x5 = 1 得基础解系 (5, -6, 0, 0, 1)^T,
方程组通解是 x = (-16, 23, 0, 0, 0)^T + k (5, -6, 0, 0, 1)^T