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sum of the matrices with fixed rank

Source: RMO district 2004,grade 12, 1

July 12, 2005
linear algebramatrixlinear algebra unsolved

Problem Statement

Let n≥2n\geq 2 and 1≤r≤n1 \leq r \leq n. Consider the set Sr=(A∈Mn(Z2),rankA=r)S_r=(A \in M_n(\mathbb{Z}_2), rankA=r). Compute the sum ∑X∈SrX\sum_{X \in S_r}X
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