For the homomorphism $h: \mathcal P_2 \rightarrow \mathcal P_2$ given by
$1 \mapsto 3$, $x \mapsto 2x-1$, $x^2 \mapsto x^2-x-1$
(i) Find the matrix $A=Rep_{\mathcal{A,A}}(h)$ for basis $\mathcal A= \langle1,x,x^2\rangle$
(ii) Find the matrix $B=Rep_{\mathcal{B,B}}(h)$ for basis $\mathcal B= \langle1,1+x,1+x+x^2\rangle$
(iii) Find the matrix $P$ such that $B=PAP^{-1}$.
For context, we are doing a unit on determinants, and this is an undergraduate linear algebra class. Here's my attempt at (i) and (ii), which is as far as I got:
(i) I start by evaluating $h(\mathcal A)$
$h(\vec a_1)=h(1)=3$
$h(\vec a_2)=h(x)=2x-1$
$h(\vec a_3)=h(x^2)=x^2-x-1$
Then, for $A=Rep_{\mathcal{A,A}}(h)$ I have:
$A = \begin{pmatrix}\\ Rep_{\mathcal A}(3) & Rep_{\mathcal A}(2x-1) & Rep_{\mathcal A}(x^2-x-1) \\ \\ \end{pmatrix}$ $= \begin{pmatrix} 3 & -1 & -1 \\ 0 & 2 & -1 \\ 0 & 0 & 1 \\ \end{pmatrix}$
Please tell me if I've calculated each respective representation right, because I fear that I've already messed up here.
(ii) Here, following the same process, I get:
$ B= \begin{pmatrix}\\ Rep_{\mathcal B}(3) & Rep_{\mathcal B}(2x+2) & Rep_{\mathcal B}(x^2+x+1) \\ \\ \end{pmatrix} = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$
Am I on the right track? Furthermore, how should I go about (iii)? I am not sure how to approach it but I assume it has to do with the properties of determinants in one way or another.
Thanks!!