The GATE Grind

GATE Linear Algebra: Eigenvalues and Canonical Forms – Previous Year Questions

20 GATE previous year questions on Eigenvalues and Canonical Forms (Linear Algebra, Mathematics) with answers and explanations, from every paper.

  1. GATE 2021 MA Q59 (2 marks, MTA) – A has characteristic polynomial det(xI−A)=x⁴(x−1)²(x−2)³. If rank(A²)<rank(A³)=rank(A⁴), find the geometric multiplicity of eigenvalue zero.
  2. GATE 2021 MA Q62 (2 marks, Numerical answer) – V is the real vector space of polynomials p=a₀+a₁x+a₂x². Define T(p)=[p(0)−p(1)]+[p(0)+p(1)]x+p(0)x². Find the sum of its eigenvalues.
  3. GATE 2022 MA Q11 (1 mark, Multiple choice) – M∈ℂ³ˣ³ has characteristic polynomial λ³+αλ²+βλ−1, with α+β≠0. Which identity holds?
  4. GATE 2022 MA Q65 (2 marks, Numerical answer) – A real 3×3 M satisfies M²=2M+3I and det M=−9. Find trace M.
  5. GATE 2023 MA Q19 (1 mark, Multiple choice) – M=pmatrix4&-3\\1&0pmatrix. P:M⁸+M¹² is diagonalizable. Q:M⁷+M⁹ is diagonalizable. Choose truth values.
  6. GATE 2023 MA Q22 (1 mark, Multiple choice) – T:ℝ⁴→ℝ⁴ has kernel 4x₁+3x₂+2x₃+x₄=0 and rank(T−3I)=3. Find its minimal polynomial.
  7. GATE 2023 MA Q35 (1 mark, Numerical answer) – Let M=pmatrix3&-1&-2\\0&2&4\\0&0&1pmatrix. If 6M⁻¹=M²−6M+αI, find α.
  8. GATE 2023 MA Q59 (2 marks, Numerical answer) – A is a real 3×3 matrix with det(A+iI)=0 and det A=3. Find trace(A²).
  9. GATE 2023 MA Q60 (2 marks, Numerical answer) – A=(a ij) is real 3×3 with A(1,2,1)ᵀ=2(1,2,1)ᵀ, A(0,1,1)ᵀ=2(0,1,1)ᵀ and A(−1,1,0)ᵀ=4(−1,1,0)ᵀ. If m is its minimal-polynomial degree, find…
  10. GATE 2024 MA Q19 (1 mark, Multiple choice) – A=pmatrix0&2\\2&0pmatrix. On M₂(ℂ), T(B)=AB. Find T’s characteristic polynomial.
  11. GATE 2024 MA Q20 (1 mark, Multiple choice) – A is a complex normal matrix. I: real eigenvalues imply A is Hermitian. II: unit-modulus eigenvalues imply A is unitary. Choose truth values.
  12. GATE 2024 MA Q51 (2 marks, Multiple select) – T,S:ℝ⁴→ℝ⁴ are nonzero and nonidentity; T²=T. Which statements hold?
  13. GATE 2024 MA Q56 (2 marks, Multiple select) – T:ℝ⁴→ℝ⁴ has only eigenvalues 1,2, dim ker(T−I)=1 and rank(T−2I)=2. Which Jordan forms are possible?
  14. GATE 2025 MA Q11 (1 mark, Multiple choice) – S is the set of unit vectors w∈ℝ³ for which A=(1,-3,2)TwT is diagonalizable. Which statement holds?
  15. GATE 2025 MA Q23 (1 mark, Multiple select) – For which invertible M is the difference of characteristic polynomials p M(x)−p (M⁻¹)(x) constant?
  16. GATE 2025 MA Q27 (1 mark, Numerical answer) – A real 7×7 M has characteristic polynomial (x−1)α(x−2)β(x−3)², α>β. Each of M−I, M−2I, M−3I has rank 5. Find m M(5), where m M is the minimal…
  17. GATE 2025 MA Q48 (2 marks, Multiple select) – T:M₂(ℝ)→M₂(ℝ) sends X to AXB, with A=pmatrix1&-2\\1&4pmatrix, B=pmatrix6&5\\-2&-1pmatrix. P is its matrix in the standard basis. Which statements…
  18. GATE 2026 MA Q28 (1 mark, Numerical answer) – T reflects ℝ³ through a two-dimensional subspace. P is its matrix in the standard basis. Find 2 trace(P) − 3 det(P) (integer).
  19. GATE 2026 MA Q48 (2 marks, Multiple select) – Nonzero real 3×3 matrices P,Q are simultaneously upper triangularizable by the same nonsingular S. Let K=PQ−QP. Which statements hold?
  20. GATE 2026 MA Q55 (2 marks, Numerical answer) – Let J=pmatrix2&1&0&0&0&0\\0&2&0&0&0&0\\0&0&2&1&0&0\\0&0&0&2&0&0\\0&0&0&0&3&1\\0&0&0&0&0&3pmatrix. Find geometric multiplicity of eigenvalue 2…