Computing the powers of matrices

[ver_mathstats]

Suppose p(λ)=(λ-1)^3 for some diagonalizable matrix A. Calculate A^25.

I'm confused as to how to approach this question without A being given. I thought perhaps I could use the characteristic equation in some way although I am still unsure. I think I could start with using λ=1. Would my matrix then be [ 1 0 0; 0 1 0; 0 0 1] then I would do [1^25 0 0; 0 1^25 0; 0 0 1^25], and finally I would arrive at my answer which would be A^25= [ 1 0 0; 0 1 0; 0 0 1]?

Thank you.

 

[member 587159]

Yes, it's correct.

You know that $P^{-1} A P = I_3$ for some matrix $P$ because $A$ can be diagonalised and has only the eigenvalue $3$. Consequently $A = I_3$ and the result follows.