GATE 2026 MA – Question 21

Linear Algebra · Inner Products and Quadratic Forms · 1 mark · Multiple select

P is real symmetric positive definite, 3×3. Which statements about $Q^TPQ$ are true?

Positive definite for every nonzero real 3×3 Q.
Positive semidefinite for every real 3×3 Q.
Positive definite when Q is nonsingular.
Not positive semidefinite for some real 3×3 Q.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) Positive semidefinite for every real 3×3 Q.; (C) Positive definite when Q is nonsingular.

Explanation

For every v, $v^TQ^TPQv=(Qv)^TP(Qv)\ge0$, proving B. If Q is invertible and v ≠ 0, Qv ≠ 0, so the expression is strictly positive, proving C. A nonzero singular Q has a nonzero kernel vector, refuting A. D contradicts B. **B,C**.

More Inner Products and Quadratic Forms questions

GATE 2026 MA Q37 – Define x,y P=xTPy on ℝ³ with P = diag(1,9,16). For V = span(1,0,0)ᵀ,(1,1,1)ᵀ, choose an…
GATE 2026 MA Q57 – Maximize z=xTpmatrix2&0&0\\0&2&1\\0&1&2pmatrixx over real vectors with xᵀx=1. Find the…
GATE 2025 MA Q61 – Under ⟨f,g⟩=∫₋₁¹f(x)g(x)dx, polynomial p(x)=α+βx²−30x⁴ is orthogonal to every polynomial…
GATE 2025 MA Q62 – M is the symmetric matrix of Q=2x₁²+2x₂²+3x₃²+4x₁x₂+2x₁x₃+2x₂x₃. For any nonzero Y∈ℝ³, a…
GATE 2024 MA Q42 – Five distinct unit vectors v₁,…,v₅ in ℝ⁷ have nonpositive integer pairwise inner products…
GATE 2023 MA Q34 – In ℝ⁴ with the standard inner product, M=x:x₁=x₃. Find dim M⊥.

All Inner Products and Quadratic Forms questions →