Question

求齐次线性方程组

Answer 1

系数矩阵行列式 |A| =
|λ+4 3 0|
|4 0 1|
|-5 λ -1|
|A| =
|λ+4 3 0|
|-1 λ 0|
|-5 λ -1|
|A| = -(λ^2+4λ+3) = -(λ+1)(λ+3)
λ = -1, 或 λ = -3 时，|A| = 0， 齐次方程组有非零解。
λ = -1 时，A =
[ 3 3 0]
[ 4 0 1]
[-5 -1 -1]
初等行变换为
[ 1 1 0]
[ 0 -4 1]
[ 0 4 -1]
初等行变换为
[ 1 1 0]
[ 0 4 -1]
[ 0 0 0]
取 x3 = 4， 则 x2 = 1， x1 = -1， 通解 x = k(-1, 1, 4)^T;
λ = -3 时，A =
[ 1 3 0]
[ 4 0 1]
[-5 -3 -1]
初等行变换为
[ 1 3 0]
[ 0 -12 1]
[ 0 12 -1]
初等行变换为
[ 1 3 0]
[ 0 12 -1]
[ 0 0 0]
取 x3 = 12， 则 x2 = 1， x1 = -3， 通解 x = c(-3, 1, 12)^T.