0
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ $Please \; don't \; use \; \LaTeX \; to \; type \; complete \; sentences.$ 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
4
$\begingroup$

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 $4$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.