GATE 2026 ST – Question 60
Y~N₃(0,I₃), A is the 3×3 matrix with every entry 1/3, and B=I₃−A. Which hold?
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Correct answer: (A) YᵀAY~χ²₁; (C) YᵀAY is independent of Y₁−2Y₂+Y₃
Explanation
A is the rank-one orthogonal projection onto span(1,1,1), so YᵀAY=(Y₁+Y₂+Y₃)²/3~χ²₁. B has rank two and its quadratic form is χ²₂. The contrast (1,−2,1) is orthogonal to the projected direction, so its normal linear form is independent of the projection and its square. The vector (1,2,1) is not orthogonal; its squared linear form has nonzero covariance with YᵀAY, ruling out independence. **A,C**.