<s>[INST] ### Calculating Transition Probabilities
Let \(P=(p_{ij})_{1\leq i,\,j\leq n}\) be a transition probability matrix for a finite-state discrete-time Markov chain. The transition probability \(p_{ij}\) denotes the probability of transitioning from state \(i\) to state \(j\).
Suppose that \(Q=(q_{ij})_{1\leq i,\,j\leq n}\) is an \(n\times n\) real matrix where each element \(q_{ij}>0\), but the elements in each row do not necessarily sum up to 1.
Can you determine a matrix \(P=(p_{ij})\) of transition probabilities such that, for \(1\leq i,\,j\leq n\),
\[p_{ij}=\frac{q_{ij}}{\sum_{k=1}^{n}q_{ik}}\text{?[/INST]