Matrix-valued function: understanding 
$A(x)=(a_{i,j}(x))_{i,j}: \mathbb{R}^n\to \mathbb{R}^{m\times k}$ is a matrix-valued function.

From my understanding it is a function that maps a vector to a matrix. How does this happen? Is the 'function' a matrix of which the elements are functions? If so, are these just $\mathbb{R}^n-\mathbb{R}$-functions? 
I don't really understand the form of a matrix-valued function and how it 'takes' the required input. Maybe an example could be helpful?
Thanks a lot.

As an example: Imagine $x=[x_1,x_2]^T\in \mathbb{R}^2$ Then the function
$$A(x)=
\begin{bmatrix}
x_1x_2 & x_1^2\sin x_2 & x_2 \\
x_1x_2^3 & x_2\exp(x_1) & 2x_2x_1\sin(x_1) \\
\end{bmatrix}$$
is a map from $\mathbb{R}^2\to \mathbb{R}^{2\times3}$. The entries of the matrix are multivariate scalar functions.
