GATE DA 2026 Set 1 — Question 52
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Linear Algebra → Matrix Algebra → Idempotent, Involutory & Nilpotent
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Question
Let be a matrix, where and is the identity matrix of order .Which of the following options is/are correct?
Correct answer
(A) M^T = M; (D) M is a projection matrix
Solution
The matrix is given by . This is the centering matrix.Option (A):
.
Thus, is symmetric. Option (A) is correct.Option (B):
Let . Note that .
.
So (unless , which implies , impossible). Option (B) is incorrect.Option (C):
.
Option (C) is incorrect.Option (D):
A matrix is a projection matrix if it is symmetric and idempotent (). Since we showed and , is a projection matrix. Option (D) is correct.
.
Thus, is symmetric. Option (A) is correct.Option (B):
Let . Note that .
.
So (unless , which implies , impossible). Option (B) is incorrect.Option (C):
.
Option (C) is incorrect.Option (D):
A matrix is a projection matrix if it is symmetric and idempotent (). Since we showed and , is a projection matrix. Option (D) is correct.
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