Discussion
Consider 4 × 4 matrices with their elements from {0, 1}. The number of such matrices with even number of 1s in every row and every column is
Answer: A
There are 512 such matrices.
Fill the top-left 3 × 3 block freely: \(2^{9} = 512\) ways. The last entry of each of the first three rows is then forced, to make that row's count of 1s even, and likewise the last entry of each of the first three columns.
The bottom-right corner is forced twice, once by the last row and once by the last column, but both conditions give the same value, because each equals the parity of the 1s in the 3 × 3 block. So every free choice extends to exactly one valid matrix, and the count is \(2^{(4-1)^{2}} = 512\).
1025, 1023 and 255 are not powers of 2, but the count must be a power of 2, since these matrices form a vector space over GF(2).
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