$\mathbf{A}$, $\mathbf{P}$ and $\mathbf{D}$ are $n \times n$ matrices.
Given above, which of the following are true and why?
A.) If $\mathbf{A}$ is diagonalizable, then $\mathbf{A}$ is invertible.
B.) If there exists a basis for $R^n$ consisting entirely of eigenvectors of $\mathbf{A}$, then $\mathbf{A}$ is diagonalizable.
C.) $\mathbf{A}$ is diagonalizable if $\mathbf{A}=\mathbf{PDP}^{-1}$ for some diagonal matrix $\mathbf{D}$ and some invertible matrix $\mathbf{P}$.
D.) $\mathbf{A}$ is diagonalizable if and only if $\mathbf{A}$ has $n$ eigenvalues, counting multiplicities.
Only B is true. I don't know why.
Hints: what does diagonalizable mean? $A$ being the zero matrix is interesting for some of these.