It is well known, though non trivial at all to prove, that a square matrix is always similar (conjugate) to its transpose, so it's enough to show that similar matrices have the same char. pol.
Assume then that , for some invertible matrix . Let us denote the determinant by |.|, so:
...end the argument now.
Here above there're quite a few non-trivial things you must know: distributivity of matrix multiplication, the fact that scalar matrices commute with any other matrix, the multiplicativity of the determinant, etc.
If you don't know these things then the above won't help much.