Question

一道线性代数题

一道线性代数题第三题。。我做a=0对么。。第二个问化出来以后那个1－矩阵怎么求啊求教谢谢

Answer 1

a = 0 对。不是 1 - 矩阵， 而应是 E - 矩阵。
A^3 = O, |A|^3 = 0, |A| = 0, a^3 = 0, a = 0.
X - XA^2 - AX + AXA^2 = E, X(E-A^2) - AX(E-A^2) = E,
(X-AX)(E-A^2) = E. (E-A)X(E-A^2) = E,
X = (E-A)^(-1)(E-A^2)^(-1) = [(E-A^2)(E-A)]^(-1)
= (E-A-A^2+A^3)^(-1) = (E-A-A^2)^(-1) (因 A^3 = O)
A =
[0 1 0]
[1 0 -1]
[0 1 0]
A^2 =
[1 0 -1]
[0 0 0]
[1 0 -1]
E-A-A^2 =
[ 0 -1 1]
[-1 1 1]
[-1 -1 2]
(E-A-A^2, E) =
[ 0 -1 1 1 0 0]
[-1 1 1 0 1 0]
[-1 -1 2 0 0 1]
初等行变换为
[ 1 -1 -1 0 -1 0]
[ 0 1 -1 -1 0 0]
[ 0 -2 1 0 -1 1]
初等行变换为
[ 1 0 -2 -1 -1 0]
[ 0 1 -1 -1 0 0]
[ 0 0 -1 -2 -1 1]
初等行变换为
[ 1 0 0 3 1 -2]
[ 0 1 0 1 1 -1]
[ 0 0 1 2 1 -1]
X = = (E-A-A^2)^(-1) =
[3 1 -2]
[1 1 -1]
[2 1 -1]