Answer
a.) False
b.) True
c.) False
d.) False
Work Step by Step
a.) False. Indeed, $A = PDP^{-1}$ is the diagonalized form of $A$, however, $D$ cannot be any vector. Matrix $D$ must be a diagonal matrix in which entries are the eigenvalues of $A$.
b.) True. From this statement, we can go ahead and build $P$ using the eigenspace basis in $ \mathbb{R}^n$.
c.) False. Every matrix will have $n$ eigenvalues (counting multiplicity). However, this does not make it diagonalizable. It must have $n$ linearly independent eigenvectors.
d.) False. 0 cannot be an eigenvalue for $A$ to be invertible. Apart from that, there is no relation between diagonalization and invertibility.
