Matrix scaling limits in finitely many iterations

The alternate row and column scaling algorithm applied to a positive $n\times n$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes called the \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a two parameter family of row but not column stochastic $n\times n$ positive matrices is constructed that become doubly stochastic after exactly one column scaling.