Is this a family of similar matrices $\left(\begin{smallmatrix} 0&x\\ 0&0 \end{smallmatrix}\right)$?

Question

Is matrix $A = \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$ similar to matrix $B=\begin{pmatrix} 0&2\\ 0&0 \end{pmatrix}$? If so, how do I prove this?

I came here from following the answer to this question:

Do similar matrices have equal singular values?

Answer 1

If they are similar then

$\begin{pmatrix} a&b\\ c&d \end{pmatrix} \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}= \begin{pmatrix} 0&2\\ 0&0 \end{pmatrix} \begin{pmatrix} a&b\\ c&d \end{pmatrix} $

for some invertible matrix $P=\begin{pmatrix} a&b\\ c&d \end{pmatrix}$.

Try to calculate such matrix $P$ from the above condition.