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GATE DA 2024 Q49: Properties of a projection matrix onto a subspace of R³

Question

Let R\mathbb{R} be the set of real numbers, UU be a subspace of R3\mathbb{R}^3 and M∈R3×3\mathbf{M} \in \mathbb{R}^{3\times3} be the matrix corresponding to the projection on to the subspace UU.

Which of the following statements is/are TRUE?

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Answer: B, C, D (official GATE 2024 final answer key)

Concept used: A projection is idempotent, and rank + nullity = 3 with rank =dim⁡U= \dim U.

First step: Projecting twice changes nothing, so M2=M\mathbf{M}^2 = \mathbf{M} (and hence M3=M\mathbf{M}^3 = \mathbf{M}); the null space has dimension 3−dim⁡U3 - \dim U.

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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