Peteryellow: Please re-read the first post. What is being asked is a necessary and sufficient condition.
ADARSH: Any matrix is similar to itself -in fact similarity is an equivalence relation-, that is what Peteryellow meant ( what I interpret, because he certainly didn't say that),
As for the question:
The 2 matrixes are similar if and only if there exists a non-singular Matrix
such that
Or equivalently:
and
is invertible
Let's work with a generic matrix:
with
(1) so that it is indeed invertible.
Now:
Equivalently:
Solve for
(see under what conditions on a, b, c, d this is possible), and then what other conditions you have to add for (1) to hold. And then you are done.