The GATE Grind

GATE Matrix Theory: Eigenvalues and Matrix Decompositions – Previous Year Questions

14 GATE previous year questions on Eigenvalues and Matrix Decompositions (Matrix Theory, Statistics) with answers and explanations, from every paper.

  1. GATE 2021 ST Q23 (1 mark, Numerical answer) – A real 2×2 matrix has eigenvalues 1,−1 with corresponding eigenvectors (√3/2,1/2)ᵀ and (−1/2,√3/2)ᵀ. If A²⁰²¹ has entries a,b,c,d,find a+b+c+d,to two…
  2. GATE 2021 ST Q40 (2 marks, Multiple choice) – For a real 3×3 A with I+A invertible,let B=(I+A)⁻¹(I−A). Which statement is true?
  3. GATE 2022 ST Q48 (2 marks, Multiple select) – A real 3×2 matrix M has SVD M=USVᵀ,where U,V are orthogonal and S=[[√3,0,0],[0,1,0]]ᵀ. Which hold?
  4. GATE 2023 ST Q12 (1 mark, Multiple choice) – A real 3×3 matrix A has eigenvalues 1,0,−1. For B=A²+2A+I₃, which identity holds?
  5. GATE 2024 ST Q48 (2 marks, Multiple select) – For a real n×n A, which hold?
  6. GATE 2024 ST Q49 (2 marks, Multiple select) – Which claims hold for real 3×3 matrices?
  7. GATE 2024 ST Q59 (2 marks, Numerical answer) – A is real 2×2 with trace 5,determinant 6. The characteristic polynomial of (A+I)⁻¹ is x²−bx+c. Find b/c (integer).
  8. GATE 2025 ST Q26 (1 mark, Numerical answer) – P=pmatrix1&2\\-1&4pmatrix and Q=P³−2P²−4P+13I₂. Find det Q (integer).
  9. GATE 2025 ST Q38 (2 marks, Multiple choice) – A 10×10 real P has diagonal entries 9/10 and off-diagonal entries −1/10. Find its rank.
  10. GATE 2025 ST Q48 (2 marks, Multiple select) – Let O be all real 3×3 orthogonal matrices with determinant 1. Which hold?
  11. GATE 2025 ST Q56 (2 marks, Numerical answer) – A 5×5 P has zeros on the diagonal, ones above and minus ones below. Find the product of its five eigenvalues (integer).
  12. GATE 2025 ST Q57 (2 marks, Numerical answer) – P=pmatrix2&1\\1&2pmatrix, Q=pmatrix1&1\\-2&4pmatrix. Find trace(P⁵+Q⁴) (integer).
  13. GATE 2026 ST Q38 (2 marks, Multiple choice) – A is real n×n,n≥2. I:every complex eigenvalue’s conjugate is also an eigenvalue. II:for a nonreal eigenvalue λ=x+iy,y≠0 and eigenvector v, Re v and Im…
  14. GATE 2026 ST Q50 (2 marks, Multiple select) – A=pmatrix 1&1&0\\0& 2&1\\0&0& 3pmatrix with real λ i. Which hold?