MathDB

Let

A=(a_{ij})

be an

m\times n

real matrix with at least one non-zero element. For each

i\in\{1,\ldots,m\}

, let

R_i=\sum_{j=1}^na_{ij}

be the sum of the

i

-th row of the matrix

A

, and for each

j\in\{1,\ldots,n\}

, let

C_j =\sum_{i=1}^ma_{ij}

be the sum of the

j

-th column of the matrix

A

. Prove that there exist indices

k\in\{1,\ldots,m\}

and

l\in\{1,\ldots,n\}

such that

a_{kl}>0,\qquad R_k\ge0,\qquad C_l\ge0,

a_{kl}<0,\qquad R_k\le0,\qquad C_l\le0.

Problem Statement