Avoiding too many steps, which is the characteristic polynomial of this matrix 7x7? And why?
(5555555555555555555555555555555555555555555555555)
Answer
As it was stated in the commentaries, the rank of this matrix is 1; so it will have 6 null eigenvalues, which means the characteristic polynomial will be in the form:
p(λ)=αλ6(λ−β)=γ6λ6+γ7λ7
Using Cayley-Hamilton:
p(A)=γ6A6+γ7A7=0
Any power of this matrix will have the same format, a positive value for all elements.
B=[1111111111111111111111111111111111111111111111111]
A=5B
A2=527B
...
A6=5675B
A7=5776B
p(A)=(γ6+35γ7)B=0⇒γ6=−35γ7
So we have: α=γ7 and β=35
p(λ)=αλ6(λ−35)
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