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

Check the true statements below:

A. $A$ is diagonalizable if and only if $A$ has $n$ eigenvalues, counting multiplicities.
B. If $A$ is diagonalizable, then $A$ is invertible.
C. $A$ is diagonalizable if $A=PDP^{-1}$ for some diagonal matrix $D$ and some invertible matrix $P$.
D. If there exists a basis for ${\mathbb R}^n$ consisting entirely of eigenvectors of $A$, then $A$ is diagonalizable.