Question

线性代数矩阵

线性代数矩阵求这三道题的具体过程，谢谢啦！

Answer 1

例3. A = 2E + B， B =
[0 1 0]
[0 0 1]
[0 0 0]
则 B^2 =
[0 0 1]
[0 0 0]
[0 0 0]
B^n = O (n ≥ 3)
A^n = (2E+B)^n
= 2^nE + n2^(n-1)EB+ [n(n+1)/2]2^(n-2)EB^2 + ...(省略号的项均为零矩阵)
= 2^nE + n2^(n-1)B + [n(n+1)/2]2^(n-2)B^2 =
[2^n n2^(n-1) n(n+1)2^(n-3)]
[ 0 2^n n2^(n-1)]
[ 0 0 2^n ]
例5. A^2 = 2A, A^3 = 4A, ...... , A^n = 2^(n-1)A
A^n - 2A^(n-1) = 2^(n-1)A - 2*2^(n-2)A = O