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Let $A$ be an $n\times n$ matrix with real entries such that the numbers in each column sum to $0$, and all diagonal entries are non-zero. So, $A$ is non-invertible, and some linear combination of columns is equal to the $\textbf{0}$ vector. Is the linear combination necessarily unique (up to constant factor)?

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Consider the following matrix, $$\begin{pmatrix} 1&-1&0&0 \\ -1&1&0&0 \\ 0&0&1&-1 \\ 0&0&-1&1 \end{pmatrix},$$ which very clearly has 2-dimensional kernel. So the answer is no, it is not necessary.

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