GATE DA 2024 Set 1 — Question 49
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Linear Algebra → Matrix Algebra → Idempotent, Involutory & Nilpotent
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Question
Let be the set of real numbers, be a subspace of and be the matrix corresponding to the projection on to the subspace .Which of the following statements is/are TRUE?
Correct answer
(B) If U is a 2-dimensional subspace of R³, then the null space of M is a 1-dimensional subspace.; (C) M² = M; (D) M³ = M
Solution
Since is a projection matrix onto subspace , it satisfies the property . This makes option (C) true.Also, . Thus, option (D) is true.By the Rank-Nullity Theorem for a matrix :For a projection matrix onto , .
- If is 1-dimensional, , so . Option (A) claims it is 1-dimensional, which is False.
- If is 2-dimensional, , so . Option (B) claims it is 1-dimensional, which is True.
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