Chain rule and multivariable differentiation Given that $F: (\mathbb{R}^3) \longrightarrow (\mathbb{R}^2)$ is defined as $F(x,y,z) = (x^2+y+z, x+y^2+xz)$, find all the linear functions defined as $g: (\mathbb{R}^2) \longrightarrow (\mathbb{R}^3)$ such that $g(0,0) = (1,1,1)$ and $(f \circ g)'(0,0)$ equals the $2 \times 2$ matrix \begin{bmatrix}1 &  1 \\ 3  & 2\end{bmatrix}
I should be solving this by using the chain rule, but I don't know how to face the problem. Any help will be appreciated.

Note that
$$
F'(x,y,z) = \pmatrix{
2x&1&1\\1&2y&x
}
$$
Now, let $g = A \pmatrix{x\\y} + b$, where $A$ is a $3 \times 2$ matrix and $b$ is a $3$-entry vector.  We have $g'(x,y) = A$, and $g(0,0) = b = (1,1,1)$, which is given.
In order to figure out what $A$ is, apply the chain rule:
$$
(f \circ g)'(0,0) = F'(g(0,0)) \cdot g'(0,0) = F'(1,1,1) \cdot A
$$
That is, we have
$$
F'(1,1,1) \cdot A = \pmatrix{1&1\\3&2}
$$
All that remains is to solve for $A$.
