$A$, $P$ and $D$ are $n\times n$ matrices.

Check the true statements below:

A. If $A$ is diagonalizable, then $A$ has $n$ distinct eigenvalues.
B. $A$ is diagonalizable if $A$ has $n$ distinct eigenvectors.
C. If $A$ is invertible, then $A$ is diagonalizable.
D. If $AP=PD$, with $D$ diagonal, then the nonzero columns of $P$ must be eigenvectors of $A$.