Let $A$ be an $n$-th order square matrix with complex entries. Which of the following statements are true?
(a) $A$ is always similar to a diagonal matrix.
(b) $A$ is always similar to an upper-triangular matrix.
(c) $A$ is similar to a block diagonal matrix, with each diagonal block of size strictly less than $n$, provided $A$ has at least $2$ distinct eigenvalues.
There are so many matrices which are not diagonalizable like nonzero nilpotent matrices. So (a) is false. I think (b) is true as when we find the rank of a matrix it is converted to a upper-triangular matrix. I have no idea about (c).
Is my thinking correct?