GATE 2026 ST – Question 51
$A=\begin{pmatrix}3&1&1\\1&0&2\\1&2&0\end{pmatrix}$ and S={x∈ℝ³:xᵀAx=1}. Which hold?
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Correct answer: (B) Every line in ℝ³ has at most two points in S.; (D) S is unbounded.
Explanation
The official key is **B,D**, retained for grading, but B has an exception that makes its printed universal claim false. A has eigenvalues 4,1,−2 and an orthogonal change of coordinates gives S:4a²+b²−2c²=1. This is nonempty and unbounded; the full line (a,b,c)=(t,1,√2t),t∈ℝ, lies in S. Transforming it back gives a line in the original S with infinitely many points, disproving B as stated. A line not wholly contained in S does have at most two intersections, which is the likely intended qualification. The solution discloses the discrepancy rather than applying that exception silently.