Question

线性代数，求解，谢谢。题目如图

Answer 1

增广矩阵 (A, b) =
[1 1 3 -1 2]
[2 1 4 -3 -1]
[3 4 -11 -2 -9]
初等行变换为
[1 1 3 -1 2]
[0 -1 -2 -1 -5]
[0 1 -20 1 -15]
初等行变换为
[1 0 1 -2 -3]
[0 1 2 1 5]
[0 0 -22 0 -20]
初等行变换为
[1 0 0 -2 -43/11]
[0 1 0 1 35/11]
[0 0 1 0 10/11]
r(A, b) = r(A) = 3 < 4, 方程组有无穷多解。
方程组化为
x1 = -43/11 + 2x4
x2 = 35/11 - x4
x3 = 10/11,
令 x4 = 0, 得特解 (-43/11, 35/11, 10/11, 0)^T;
导出组是
x1 = 2x4
x2 = -x4
x3 = 0,
令 x4 = 1, 得 Ax = 0 的基础解系 (2, -1, 0, 1)^T;
方程组的通解是 x = k (2, -1, 0, 1)^T + (-43/11, 35/11, 10/11, 0)^T。