Hi can anyone help consolidate my partial answers to parts d) e) f), as I think I've gone wrong somewhere but do not know where.
d) firstly I found out what the matrix of $C$ looks like in base $B$:
$$\begin{align}[C]_B &= ([x^2-x]_B|[x]_B|[-x^2+x+1]_B)\\ &=\begin{bmatrix}0 & 0 & 1\\ -1 & 1 & 1\\1 & 0 & -1\end{bmatrix}\end{align}$$
So it is implied that $[C]_B [B]_C = $ Identity Matrix
So $[B]_C=[C]_B^{-1}$
Therefore $$[B]c= \begin{bmatrix}1 & 0 & 1\\ 0 & 1 & 1\\1 & 0 & 0\end{bmatrix}$$
Now for e)
I have first found $T$ in the basis $B$:
$$\begin{align}[T]_B &= ([T(1)]_B|[T(x)]_B|[T(x^2)]_B)\\ &= \begin{bmatrix}-1 & 0 & 0\\-2 & 2 & 1\\2 & 0 & 1\end{bmatrix}\end{align}$$
Next I saw what $C$ looks like in the basis of $B$:
$$\begin{align}[C]_B &= ([x^2-x]_B|[x]_B|[-x^2+x+1]_B)\\ &= \begin{bmatrix} 0 & 0 & 1\\-1 & 1 & 1\\1 & 0 & -1\end{bmatrix}\end{align}$$
Now I computed the inverse of $C$ : $$\begin{bmatrix} 1 & 0 & 1\\0 & 1 & 1\\1 & 0 & 0\end{bmatrix}$$
And have checked this is correct as $CC^{-1}$ is the identity matrix.
Then I proceeded to compute $[T]_C$:
$$P^{-1}[T]_BP = \begin{bmatrix} 1 & 0 & 0\\0 & 2 & 0\\0 & 0 & -1\end{bmatrix}$$
And here is where I get confused. As I suspected from being such a small part of the question, the generalisation of $[T]_C^N$ would have come from $[T]_C$ being idempotent and thus $N$ could be any integer but this is not the case or I've made a mistake somewhere.
So instead I've wrote:
$$[T]_C^N = \begin{bmatrix} 1^N & 0 & 0\\0 & 2^N & 0\\0 & 0 & -1^N\end{bmatrix}$$
Can anyone advise me one this. Much appreciated.