Possibility of arranging $1$ and $-1$ in a grid such that the sum of the products is $0$ Consider an $11$ x $11$ grid, where in each square, the number $1$ or $-1$ is written. One multiplies the numbers in each row and column, and then sums up these $22$ products. Is it possible for this sum to be equal to $0$?
My approach was to realize that the product in every row or column is either $1$ or $-1$. This means, since we have $22$ products, that $11$ of them must be equal to $1$ and the other $11$ to $-1$. If there is an odd number of $-1$:s in a row or column, the product will be $-1$, while if there is an even number of $-1$:s, the product will be $1$. Then, I considered coloring each square in the following way: black if the cell contains a $-1$, uncolored if the cell contains a $1$. Thus, the question becomes: can one color grid cells in an $11$ x $11$ grid in such a way that an odd number of cells is colored in $11$ rows/columns, and an even number of cells is colored in the remaining $11$ rows/columns? However, from here, I couldn't make much progress. Does anyone have any ideas?

Start with a grid $11\times11$, with $1$ in each square. Total is $22$.
Change $1$ to $-1$ in one square, new total is $18$.
Start with a grid $11\times11$, with random values. Change $1$ value, new total of rows increase or decrease by $2$, new total of columns increase or decrease by $2$, new total of rows+columns change by $-4, 0$ or $+4$.
Total is always equal to $2\pmod 4$. Total cannot be $0$, neither $4$ or $8$ or $12$ ...
