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GATE DA 2025 Q13: Row sums equal 1: solutions of Bx = 0 for B = A³ − 2A² + A

Question

The sum of the elements in each row of A∈Rn×nA \in \mathbb{R}^{n \times n} is 1. If B=A3−2A2+AB = A^3 - 2A^2 + A, which one of the following statements is correct (for x∈Rnx \in \mathbb{R}^n)?

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Answer: C (official GATE 2025 final answer key)

Concept used: Row sums of 1 mean the all-ones vector is an eigenvector with eigenvalue 1; a non-trivial null space gives infinitely many solutions.

First step: Factor B=A(A−I)2B = A(A - I)^2 and let 1\mathbf{1} be the all-ones vector. Since A1=1A\mathbf{1} = \mathbf{1}, check what BB does to 1\mathbf{1}.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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