Question: Currency exchange matrix. We consider a set of \(n\) currencies, labeled \(1, \ldots, n\). (These might correspond to USD, RMB, EUR, and so on.) At a particular time the exchange or conversion rates among the \(n\) currencies are given by an \(n \times n\) (exchange rate) matrix \(R\), where \(R_{i j}\) is the amount of currency \(i\) that you can buy for one unit of currency \(j\). (All entries of \(R\) are positive.) The exchange rates include commission charges, so we have \(R_{j i} R_{i j}<1\) for all \(i \neq j\). You can assume that \(R_{i i}=1\). Suppose \(y=R x\), where \(x\) is a vector (with nonnegative entries) that represents the amounts of the currencies that we hold. What is \(y_{i}\) ?

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