Question

求线性方程组{+x1+2x2+x3-2x3=3+2x1+3x2-x4=5+x1-x2-5x3+7x

Answer 1

(4) 增广矩阵 (A, b) =
[1 2 3 -1 2]
[3 2 -1 -1 4]
[1 -2 -5 -1 0]
初等行变换为
[1 2 3 -1 2]
[0 -4 -10 2 -2]
[0 -4 -8 0 -2]
初等行变换为
[1 2 3 -1 2]
[0 -4 -10 2 -2]
[0 0 2 -2 0]
初等行变换为
[1 0 -2 0 1]
[0 -2 -5 1 -1]
[0 0 2 -2 0]
初等行变换为
[1 0 0 -2 1]
[0 -2 0 -4 -1]
[0 0 1 -1 0]
初等行变换为
[1 0 0 -2 1]
[0 1 0 2 1/2]
[0 0 1 -1 0]
取 x4 = c， 方程组的解为
x1 = -2c + 1
x2 = 2c + 1/2
x3 = -c
x4 = c