Problem
COM-B1-M06-P021 One Negative Cell
#21
★★★★☆ Level 4 of 5
In a \(6\) by \(6\) table, all entries are \(1\). In one move, one may change signs of all entries in one chosen row or column. Can one obtain a table with exactly one entry \(-1\) and all others \(1\)?
Each move changes \(6\) signs.
The product of all \(36\) entries is preserved: one move multiplies it by \((-1)^6=1\). Initially the product is \(1\). A table with one \(-1\) has product \(-1\). Thus it is unreachable.
Same idea on a larger board.