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Consider the following square matrix \begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} & a_{1,4} \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} \\ a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4} \\ a_{4,1} & a_{4,2} & a_{4,3} & a_{4,4} \end{bmatrix} where $0\leq a_{i,j} \leq 1$ for all $i,j \in [1,16]$. Consider some $\delta >0$, I know that this matrix satisfies \begin{align} \left| \sum_{j=1}^4 a_{i,j} - \frac{1}{4} \sum_{i=1}^4 \sum_{j=1}^4 a_{i,j} \right| \leq \delta \quad \text{for all } i \in [1,4], \end{align} that is, all the row sums almost equal to the sum of all elements divided by the number of rows (or columns).

QUESTION: The previous property implies \begin{align} \left| \sum_{i=1}^4 a_{i,j} - \frac{1}{4} \sum_{i=1}^4 \sum_{j=1}^4 a_{i,j} \right| \leq \delta \quad \text{for all } j \in [1,4], \end{align} ?????????

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No, there are easy counter examples: $$ \begin{bmatrix} 1&1&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}$$

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