As elegant as @ki3i's solution is, what if we weren't lucky enough to see the easy way out?
We'd have to start from the constraint given by the determinant that $p+q+r=2+pqr$ and make it look like the desired expression.
then, by multiplying out and distributing, etc, we'll get that:
Substituting that $pqr=p+q+r-2$, we get:
This equation can then be factored:
Here, we can clearly see that if $k=1$ we're done. So it's clear that the expression can evaluate to one.
To finish things off and show that it must evaluate to 1, we should show that the second factor is 0 and the determinant constraint holds $\Leftrightarrow$ (at least) one of $p,q,r$ is 1.
(completed by @ki3i) Since
$p=1$ or $q=1$ or $r=1$, this completes the proof.