Consider this question asked in my masters entrance for which I am preparing.
Let A be a real matrix with characterstic polynomial $(X-1)^3$ . Choose correct statements:
(1) A is always diagonazable.
(2) If minimal polynomial of A is $(X-1)^3$ then A is diagonizable,
(3) Characterstic polynomial of $A^2$ is $(X-1)^3$ .
(4) If A has exactly 2 jordan blocks , then $(A-1)^2$ is diagonizable.
I have proved a false , c true . I don't know how to approach (b),(d)( which concept should be used?).
Kindly give a detailed answer for (D) as i am not good in Jordan Forms
If useful ,I followed Hoffman Kunze
(2) is false. In fact, a matrix is diagonalizable iff its minimal polynomial has no repeated factors.
(4) is true. The key is that for a Jordan block $J$ associated with $\lambda$, the size of $J$ is equal to the smallest $k$ for which $(J- \lambda)^k = 0$. If there are exactly two Jordan blocks for a size $3$ matrix, then each block has size $2$ or smaller.