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GATE DA 2026 Q52: Properties of the centering matrix

Question

Let M=(In−1n11T)M = \left(I_n - \dfrac{1}{n}\mathbf{1}\mathbf{1}^T\right) be a matrix, where 1=(1,1,1,…,1)T∈Rn\mathbf{1} = (1, 1, 1, \ldots, 1)^T \in \mathbb{R}^n and InI_n is the identity matrix of order nn.

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Answer: A, D (official GATE 2026 final answer key)

Concept used: 1n11T\tfrac{1}{n}\mathbf{1}\mathbf{1}^T projects onto the all-ones direction; MM projects onto its orthogonal complement.

First step: Use 1T1=n\mathbf{1}^T\mathbf{1} = n to expand M2=In−2n11T+1n11T=MM^2 = I_n - \tfrac{2}{n}\mathbf{1}\mathbf{1}^T + \tfrac{1}{n}\mathbf{1}\mathbf{1}^T = M; the trace is n−1n - 1.

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

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