A Positive 3x3 matrix that is Row Stochastic, but not Column Stochastic, and becomes Doubly Stochastic after one Column Scaling By Shalosh B. Ekhad Mel Nathanson asked in his JMM 2019 talk, and also as problem 1 in "Alternat\ e Minomization and Doubly Stochastic Matrices" arXiv:1812.119302v2 and as problem 5 in his paper "Matrix Scaling, Explicit\ Sinkhorn limits, and Arithmetic" whether there exists a positive 3x3 matrix that is row stochastic, but not \ column stochastic, and becomes doubly stochastic after one column scalin\ g Here I answer the question in the affirmative. Indeed the matrix [1/5 1/5 3/5] [ ] [2/5 1/5 2/5] [ ] [3/5 1/5 1/5] and in LaTex \left[ \begin {array}{ccc} 1/5&1/5&3/5\\ \noalign{\medskip}2/5&1/5&2/5 \\ \noalign{\medskip}3/5&1/5&1/5\end {array} \right] is is row stochastic, but not column stochastic Applying column scaling, we get the doubly-stochastic matrix [1/6 1/3 1/2] [ ] [1/3 1/3 1/3] [ ] [1/2 1/3 1/6] and in LaTex \left[ \begin {array}{ccc} 1/6&1/3&1/2\\ \noalign{\medskip}1/3&1/3&1/3 \\ \noalign{\medskip}1/2&1/3&1/6\end {array} \right] By multiplying the , 3, rows, respectively, by,, 5, 5, 5, we get a matrix with integer coefficients, we get the matrix [[1, 1, 3], [2, 1, 2], [3, 1, 1]] and in LaTex [[1,1,3],[2,1,2],[3,1,1]] this becomes doubly-stochastic after TWO applications of the Sinkhorn proces\ s (or one double-application [[1, 1, 3], [2, 1, 2], [3, 1, 1]] A Positive 3x3 matrix that is Row Stochastic, but not Column Stochastic, and\ becomes Doubly Stochastic after one Column Scaling By Shalosh B. Ekhad Mel Nathanson asked in his JMM 2019 talk, and also as problem 1 in "Alternat\ e Minomization and Doubly Stochastic Matrices" arXiv:1812.119302v2 and as problem 5 in his paper "Matrix Scaling, Explicit\ Sinkhorn limits, and Arithmetic" whether there exists a positive 3x3 matrix that is row stochastic, but not \ column stochastic, and becomes doubly stochastic after one column scalin\ g Here I answer the question in the affirmative. Indeed the matrix [ 17 ] [1/6 1/8 -- ] [ 24 ] [ ] [1/3 3/8 7/24] [ ] [1/2 1/2 0 ] and in LaTex \left[ \begin {array}{ccc} 1/6&1/8&{\frac {17}{24}} \\ \noalign{\medskip}1/3&3/8&{\frac {7}{24}}\\ \noalign{\medskip}1/2&1/ 2&0\end {array} \right] is is row stochastic, but not column stochastic Applying column scaling, we get the doubly-stochastic matrix [ 17 ] [1/6 1/8 -- ] [ 24 ] [ ] [1/3 3/8 7/24] [ ] [1/2 1/2 0 ] and in LaTex \left[ \begin {array}{ccc} 1/6&1/8&{\frac {17}{24}} \\ \noalign{\medskip}1/3&3/8&{\frac {7}{24}}\\ \noalign{\medskip}1/2&1/ 2&0\end {array} \right] By multiplying the , 3, rows, respectively, by,, 24, 24, 2, we get a matrix with integer coefficients, we get the matrix [[4, 3, 17], [8, 9, 7], [1, 1, 0]] and in LaTex [[4,3,17],[8,9,7],[1,1,0]] NU this becomes doubly-stochastic after TWO applications of the Sinkhorn proces\ s (or one double-application [[4, 3, 17], [8, 9, 7], [1, 1, 0]]