GATE CS 2026 Set 1 — Question 12
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Engineering Mathematics → Linear Algebra → Matrices & Determinants
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Question
Consider matrices with their elements from . The number of such matrices with even number of 1s in every row and every column is
Correct answer
(A) 512
Solution
We are looking for the number of binary matrices (where ) such that every row sum and every column sum is even (i.e., ).Let the matrix be . The condition implies that for each row , , and for each column , .Consider the submatrix of size formed by the first rows and columns. We can fill this submatrix arbitrarily with s and s. There are entries, so there are ways to do this.Once the submatrix is fixed:
For , the number is .
1.The last element of each of the first rows is uniquely determined to satisfy the row parity constraint.
2.The last element of each of the first columns is uniquely determined to satisfy the column parity constraint.
3.The element at position is determined by the parity of the last row. We must check if this choice also satisfies the parity of the last column.
- Sum of all elements in the matrix = .
- Since the first row sums are even, and the last row sum is forced to be even by the choice of , the total sum is even.
- Since the first col sums are even, the last col sum must be (Total Sum - sum of first cols) = Even - Even = Even. Thus, the constraint is automatically satisfied.
For , the number is .
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