Question

大一线性代数矩阵题，求大佬解答，22题

Answer 1

AP = P∧, A = P∧P^(-1)
(P, E) =
[1 1 1 1 0 0]
[1 0 -2 0 1 0]
[1 -1 1 0 0 1]
初等行变换为
[1 1 1 1 0 0]
[0 -1 -3 -1 1 0]
[0 -2 0 -1 0 1]
初等行变换为
[1 0 -2 0 1 0]
[0 1 3 1 -1 0]
[0 0 6 1 -2 1]
初等行变换为
[1 0 0 1/3 1/3 1/3]
[0 1 0 1/2 0 -1/2]
[0 0 1 1/6 -1/3 1/6]
P^(-1) =
[1/3 1/3 1/3]
[1/2 0 -1/2]
[1/6 -1/3 1/6]
A^n = P∧P^(-1)P∧P^(-1)P∧P^(-1)......P∧P^(-1)P∧P^(-1) = P∧^nP^(-1)

φ(A) = A^8(5E-6A+A^2) = 5A^8-6A^9+A^10
= 5P∧^8P^(-1) - 6P∧^9P^(-1) + P∧^10P^(-1)
= P∧^8(5E-6∧+∧^2)P^(-1)
= Pdiag(1, 1, 5^8)diag(12, 0, 0)P^(-1) = Pdiag(12, 0, 0)P^(-1) =
[4 4 4]
[4 4 4]
[4 4 4]