GATE Matrix Theory: Inner Products and Quadratic Forms – Previous Year Questions
10 GATE previous year questions on Inner Products and Quadratic Forms (Matrix Theory, Statistics) with answers and explanations, from every paper.
- GATE 2021 ST Q39 – Let M be all symmetric positive definite real 3×3 matrices,and S=A∈M:A⁵⁰−A⁴⁸/4=0. How many elements are in S?
- GATE 2022 ST Q23 – A unit vector u∈ℝ⁴ is orthogonal to S=x:x₁+x₂+x₃+x₄=0. Let v=(1,−2,1,1)ᵀ. If u and v−αu are orthogonal,find α² to two decimals.
- GATE 2022 ST Q36 – A symmetric 3×3 matrix M has eigenvalues −1,1,2 and corresponding unit eigenvectors u,v,w. If Mx=u+2(v+w) and M²y=u−(v+2w),find x+y ².
- GATE 2022 ST Q47 – M is symmetric 3×3 with eigenvalues 1,2,3. N has real eigenvalues and MN+NᵀM=3I. Which cannot be eigenvalues of N?
- GATE 2023 ST Q36 – For a real n×n matrix A: I:if A is symmetric,there exists c≥0 for which A+cI is symmetric positive definite. II:if A is symmetric positive…
- GATE 2023 ST Q59 – In ℝ³ with standard inner product,the orthonormal basis is v₁=(1,−1,1)ᵀ/√3,v₂=(1,2,1)ᵀ/√6,v₃=(1,0,−1)ᵀ/√2. For u=(a,b,c)ᵀ,its inner products with…
- GATE 2024 ST Q15 – In standard ℝ², u=(a,b)ᵀ has inner products 2 with (1,2)ᵀ and −1 with (4,−2)ᵀ. Which holds?
- GATE 2025 ST Q13 – Let S=(x,y,z)∈ℝ³∖0:z=−(x+y). Define S⊥=v∈ℝ³:v·s=0 for every s∈S. Which holds?
- GATE 2026 ST Q26 – In ℝ⁴ let U=x:x₁+x₂+x₄=0,x₃=0. Find dim U⊥ (integer).
- GATE 2026 ST Q51 – A=pmatrix3&1&1\\1&0&2\\1&2&0pmatrix and S=x∈ℝ³:xᵀAx=1. Which hold?