A. False. Consider the identity matrix, which is diagonalizable (it's already diagonal) but all its eigenvalues are the same (1).
b. True. Suppose
is the matrix of the eigenvectors of
, and
is the diagonal matrix of the eigenvalues of
:
Then
In other words, the columns of
are
, which are identically
, and these are the columns of
.
c. False. A counterexample is the matrix
which is nonsingular, but it has only one eigenvalue.
d. False. Consider the matrix
with eigenvalue
and eigenvector
, where
. But the matrix can't be diagonalized.