We are given the following:
Suppose $f: \mathbb{R}^2 \to \mathbb{R}^2$ and $g:\mathbb{R}^2 \to \mathbb{R}^2$ are differentiable. Let $a \in \mathbb{R}^2$ and $b = f(a)$. If $\nabla f_1(a)=(2,-1)$ , $\nabla f_2(a)=(-3,-1)$ , $\nabla g_1(b)=(-1,0)$ , $\nabla g_2(b)=(1,-3)$, then find $J(g \circ f)$.
Currently, I have reasoned, and correct me if I'm wrong, that the Jacobian Matrix will be as follows: $$\begin{pmatrix}-1 & 0\\\ 1 & -3\end{pmatrix}$$
I solved this by just using the gradient of g, which seems rather simple, so I'm not confident. Does anyone have a hint if this is indeed incorrect?