几个线性代数的题,

几个线性代数的题,
几个线性代数的题,

几个线性代数的题,
1. 记 XA=B, 则 X=BA^(-1)
(A, E) =
[1 -1 1 1 0 0]
[1 1 0 0 1 0]
[2 1 1 0 0 1]
行初等变换为
[1 -1 1 1 0 0]
[0 2 -1 -1 1 0]
[0 3 -1 -2 0 1]
行初等变换为
[1 -1 1 1 0 0]
[0 2 -1 -1 1 0]
[0 6 -2 -4 0 2]
行初等变换为
[1 -1 1 1 0 0]
[0 2 -1 -1 1 0]
[0 0 1 -1 -3 2]
行初等变换为
[1 -1 0 2 3 -2]
[0 2 0 -2 -2 2]
[0 0 1 -1 -3 2]
行初等变换为
[1 0 0 1 2 -1]
[0 1 0 -1 -1 1]
[0 0 1 -1 -3 2]
则 A^(-1) =
[ 1 2 -1]
[-1 -1 1]
[-1 -3 2]
X = BA^(-1) =
[ 2 9 -5]
[-2 -8 6]
[-6 -16 11]

2. |λE-A| =
|λ+2 -1 -1|
| 0 λ-2 0|
| 4 -1 λ-3|
=(λ-2)[(λ+2)(λ-3)+4] = (λ+1)(λ-2)^2.
A 的特征值是 λ=-1,2,2.
对于特征值λ=-1,λE-A =
[1 -1 -1]
[0 -3 0]
[4 -1 -4]
行初等变换为
[1 -1 -1]
[0 1 0]
[0 0 0]
得特征向量为 (1, 0, 1)^T;
对于重特征值λ=2,λE-A =
[4 -1 -1]
[0 0 0]
[4 -1 -1]
行初等变换为
[4 -1 -1]
[0 0 0]
[0 0 0]
得特征向量为 (1,4, 0)^T,(1, 0, 4)^T.
有两个线性无关的特征向量,
则A可以相似于对角阵 B=diag(-1, 2, 2).

3. A = (α1,α2,α3, α4) =
[ 1 2 0 3]
[ 2 0 -4 -2]
[-1 3 5 7]
[ 1 0 -2 -1]
行初等变换为
[ 1 2 0 3]
[ 0 -4 -4 -8]
[ 0 5 5 10]
[ 0 -2 -2 -4]
行初等变换为
[ 1 0 -2 -1]
[ 0 1 1 2]
[ 0 0 0 0]
[ 0 0 0 0]
其一个极大线性无关组是 α1, α2.
α3=-2α1+α2, α4=-α1+2α2.
4. 增广矩阵 （A,b）=
[1 2 -1 4 2]
[1 -3 2 -3 -1]
[1 7 -4 11 5]
行初等变换为
[1 2 -1 4 2]
[0 -5 3 -7 -3]
[0 5 -3 7 3]
行初等变换为
[1 2 -1 4 2]
[0 5 -3 7 3]
[0 0 0 0 0]
r(A)=r(A,b)=2