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GATE CSE 2026 Session 1 Q12: 4×4 binary matrices with even parity in every row and column

Question

Consider 4×44 \times 4 matrices with their elements from {0,1}\{0, 1\}. The number of such matrices with even number of 1s in every row and every column is

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Answer: A (official GATE 2026 final answer key)

Concept used: Parity constraints: the top-left (n−1)×(n−1)(n-1) \times (n-1) block is free and determines the rest.

First step: Choose the top-left 3×33 \times 3 block freely (292^9 ways) and check the last row and column are then forced consistently.

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Source: official GATE 2026 Session 1 CS question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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