Question

求解此线性代数题

Answer 1

因 r(A) ≥ 1，r(B) ≥ 2， AB = O， 只有 r(A) = 1，r(B) = 2, 得
(1) a = -1, c = -2, b = 2.
(2) r(A) = 1, A 有二重特征值 0，A 的非零特征值 λ = tr(A) = 6.
(3) 对于 λ = 6， λE - A = 6E - A =
[ 5 1 -2]
[ 1 5 2]
[-2 2 2]
初等行变换为
[-1 1 1]
[ 0 6 3]
[ 0 6 3]
初等行变换为
[-1 -1 0]
[ 0 2 1]
[ 0 0 0]
得基础解系即特征向量为(1, -1, 2)^T,
对于 λ = 0， λE - A = - A =
[-1 1 -2]
[ 1 -1 2]
[-2 2 -4]
初等行变换为
[ 1 -1 2]
[ 0 0 0]
[ 0 0 0]
得基础解系即特征向量为(1, 1, 0)^T, (2, 0, -1)^T
取 P =
[ 1 1 2]
[-1 1 0]
[ 2 0 1]
得 P^(-1)AP = diag(6, 0, 0)