which of the statements are true for a 5*5 matrix whose all entries are 1 ?

Question

1. A is not diagonalizable

2.The minimal polynomial and the characteristic polynomial of A are not equal

Answer 1

AA is symmetric (or selfadjoint, if your matrices are complex), so it is diagonalizable.

 

The minimal polynomial of A is pm(t)=t(t−5) pm(t)=t(t−5), while the characteristic polynomial is

pc(t)=t^(4)(t−5). So they are different

 

 

The eigenvalues are then 5,0,0,0,0, and so pc(t)=t^(4)(t−5) . As A^(5)−5A=0 the minimal polynomial is at most t(t−5). But it also requires 0 and 5 as roots, so pm(t)=t(t−5).

Comments

https://math.stackexchange.com/questions/238999/a-is-5%C3%975-matrix-all-of-whose-entries-are-1

can someone explain how in this they are calculating characteristic  polynomial of such huge matrix