Vtyjkyuk

This might sound quite trivial. Let $mathbfx = beginpmatrixx_1,ldots,x_N endpmatrix^top$ which takes on values on the $N$-dimensional orthant $mathcalO_N = [0,infty)^N$. If $mathbfy$ is linear transformation of $mathbfx$ defined as $mathbfy = mathbfAmathbfx$ where $mathbfA$ is an $Ntimes N$ transformation matrix (i.e., invertible),

$mathbf1.$ is the image still $mathcalO_N$?

$mathbf2.$ If no, is there a necessary and sufficient condition in $mathbfA$ for the the image to be preserved?

Context: Currently working on multivariate integration involving change-of-variables resulting to change in domain of integration.

No, in general the image is not $mathcalO_N$. This is easy to see, for example take $N=2$ and $A$ to be the standard rotation matrix:
$$beginbmatrix
costheta & -sintheta \
sintheta & costheta \
endbmatrix
$$

Here, if you put $theta = 90^circ$, then the image is completely disjoint from $mathcalO_N$. Or you can use the shear matrix to obtain situations where the image is strictly contained in $mathcalO_N$ or will strictly contain $mathcalO_N$.

For your second question, it is easy to detect if the image is contained in $mathcalO_N$. To check that, we just need to check if all the basis vectors of $mathcalO_N$ land in $mathcalO_N$.

To check if the image is exactly $mathcalO_N$, we'll have to check whether every element of $mathcalO_N$ (or at least the basis) has a preimage which will be no different than looking at the image of $mathcalO_N$ and comparing.