# Column entries of a matrix sum to zero, so what are the properties?

What kind of properties does a matrix whose column entries sum to zero have?

$$\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}$$

Where $a_{11}+\cdots+a_{m1}=0$ and so on.

You have $$a_{mk} = - \sum_{j=1}^{m-1} a_{jk}$$ for all $k$. This means the last row vector is a linear combination of the remaining row vectors. Hence, the rank of the matrix is at most $\min(m,n)-1$.
• @user17762 Don't you mean $m-1$? – Trancot May 22 '13 at 19:12
• @Trancot Nope, it is $\min(m,n)-1$. For instance, consider the matrix $$\begin{bmatrix}1 & -1\\ 2 & -2\\ 3 & -3\\ 4 & -4 \end{bmatrix}$$ Here $m=4$ and $n=2$. But rank is $n-1=1$. – user17762 May 22 '13 at 19:58
• @user17762, Oh, no, I mean the upper index on your sum, namely $\overset{\overset{?}{n-1}}{\sum}$. – Trancot May 22 '13 at 20:26