There is a matrix $A$ which has $19$ rows and $19$ columns. Every element has a value either $1$ or $-1$. Defined that $R_k$ is the product of every elements in the $k^{th}$ row and $C_k$ is the product of every element in the $k^{th}$ column. Prove that $$\sum_{i=1}^{19}(R_i+C_i)\neq0$$

help me please, this is quite hard

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    $\begingroup$ $Please \; don't \; use \; \LaTeX \; to \; type \; complete \; sentences.$ It looks much better this way. $\endgroup$ – user642796 May 31 '13 at 9:02
  • $\begingroup$ Sorry, i'm new here, thank you for editing $\endgroup$ – Jake Timberwood May 31 '13 at 9:04
  • $\begingroup$ No worries; it often takes new users a while to figure out when/where to properly use the $\LaTeX$ functionality. $\endgroup$ – user642796 May 31 '13 at 9:11

Hint: Let's first consider a matrix $A$ consisting only of $1$s. Then $C_i = R_i = 1$ for each $i$, so your sum equals 38. Whenever you "flip" a 1 to replace it with a $-1$, exactly one $C_i$ and one $R_i$ flip, that is the sum is changed by $4$, $0$ or $-4$. But $38$ is no multiple of $4$.


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