Given A is a $7\times7$ matrix with characteristic polynomial $ (\lambda -3)^5 (\lambda-1)^2$ and if minimal polynomial of A is irreducible, is A diagonalizable?
My attempt: I am not sure of what would be the possibilities for minimal polynomial to be irreducible. If minimal polynomial is $(\lambda-1)$ or $(\lambda-3)$, then it would be irreducible, but then minimal polynomial should contain all eigen values making $(\lambda-1) (\lambda-3)$ as the only possibility. Please suggest.