For which numbers $c$ and $d$ does the following matrix have rank $2,$ and rank $1?$

$A=\left[\begin{array}{ccccc}1& -1& 2& 0& 5\\ 0& 0& c& 3& 3\\ 0& 0& 0& d& 3\end{array}\right]$Let $A=\left[\begin{array}{cc}4& 0\\ 1& 2\end{array}\right].$

$($a$)$ Find the eigenvectors and the eigenvalues of $A.$

$($b$)$ Find a matrix $P$ that diagonalize $A.$

$($c$)$ Use part $($a$)$ to find $A^{15}.$