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.
[Math] Column entries of a matrix sum to zero, so what are the properties
abstract-algebradeterminantlinear algebramatrices
Best Answer
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 $\min(m,n)-1$.