In how many ways can one fill a m×n table with ±1n such that the product of the entries in each row and each column equals-1?
A
Pm−1∏i=1ain=(−1)m−1 hence (∗) holds if and only if m and n have the same parity.
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B
Pm∏i=1ain=(−1)m−1 hence (∗) holds if and only if m and n have the same parity.
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C
Pm−1∏i=1ain=(1)m−1 hence (∗) holds if and only if m and n have the same parity.
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D
Pm+1∏i=1ain=(−1)m−1 hence (∗) holds if and only if m and n have the same parity.
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Solution
The correct option is APm−1∏i=1ain=(−1)m−1 hence (∗) holds if and only if m and n have the same parity. Denote by aij=±1 in an arbitrary way. This can be done in 2(m−1)(n−1) The values for amj with 1≤j≤n−1 and for ain with 1≤i≤m−1 are uniquely determined by the condition that the product of the entries in each row and each column equals −1. The value of a amn is also uniquely determined but it is necessary that ∏n−1j=1amj=∏m−1i=1ain. If we denote P=∏m−1i=1∏n−1j=1aij we observe that Pn−1∏j=1amj=(−1)n−1 and Pm−1∏i=1ain=(−1)m−1 hence ( ) holds if and only if m and n have the same parity.