How to do integration with non-square matrix?

I am not sure how to do the following integration with a non-square matrix $A$. By Google search I could get results only for square matrix. But for non-square matrix $\det(A)$ does not exist. So how to do the following?

$\int Ax \: dx$ = ?, where $A$ has dimension $n \times p$ and $x$ has dimension $p \times 1$

• It's essentially a column vector of dimension $n\times 1$ , maybe you gotta integrate it like that of a vector? – Mann May 13 '15 at 16:57
• "$dx$" is the volume element in $\mathbb{R}^p.$ Also, $Ax$ is a vector valued function, i.e. $F:\mathbb{R}^p \to \mathbb{R}^n,$ defined by $F(x)=Ax.$ The only way this makes to me is to integrate each components of $Ax$ over some subset of $\mathbb{R}^p.$ – matt biesecker May 13 '15 at 17:01
• You forgot the domain of integration, without which your integral probabli is not finite $\frown$. Also, multiplying by a matrix is a liear transformation, so just move $A$ out of the integral. – kjetil b halvorsen May 13 '15 at 17:09