Let $T : \mathbb{P}^2 \rightarrow \mathbb{P}^2$ be the linear transformation given by $(Tp)(x) = p(x) + (x-1)p'(x)$ for $p \in \mathbb{P}^2$. Find the matrix for $T$ relative to the standard basis.
I'm assuming that because it's $\mathbb{P}^2$ that the matrix should be a 3 x 3, but I'm not sure if I got it right. So far I have: $$T(1, 0, 0) = 1 + (x-1)(0) = 1 $$ $$T(0, 1, 0) = x + (x-1)(1) = 2x - 1$$ $$T(0, 0, 1) = x^2 + (x-1)(2x) = 3x^2 - 2x$$
And so the matrix I have for T is $$\begin{bmatrix}1 & 0 & 0\\-1 & 2 & 0\\0 & -2 & 3\end{bmatrix}$$
Is this correct, and if not then where did I go wrong?