The GATE Grind

GATE Matrix Theory: Vector Spaces and Linear Systems – Previous Year Questions

15 GATE previous year questions on Vector Spaces and Linear Systems (Matrix Theory, Statistics) with answers and explanations, from every paper.

  1. GATE 2021 ST Q22 (1 mark, Numerical answer) – A has rows (0,0,1),(1,0,0),(0,1,0). Find the nullity of 5A(I₃+A+A²).
  2. GATE 2021 ST Q56 (2 marks, Numerical answer) – Let A=[a u₁ u₂ u₃],B=[b u₁ u₂ u₃],C=[u₂ u₃ u₁ a+b] be 4×4 matrices of column vectors. If det A=6,det B=2,find det(A+B)−det C.
  3. GATE 2022 ST Q11 (1 mark, Multiple choice) – A real 2×2 matrix M satisfies (I+M)⁻¹=I−αM with α≠0 and tr M=3. Find α.
  4. GATE 2022 ST Q22 (1 mark, Numerical answer) – A square matrix M of order n satisfies M²=0 and nullity(M)=6. Find the maximum possible n as an integer.
  5. GATE 2023 ST Q13 (1 mark, Multiple choice) – For real n×n matrices: I:if B is invertible,rank(BA)=rank(A). II:if A²x=b is solvable for every b∈ℝⁿ,then Ax=b is solvable for every b. Which hold?
  6. GATE 2023 ST Q30 (1 mark, Numerical answer) – A real 2×2 matrix A commutes with every real 2×2 matrix B. If tr A=5,find det A to two decimals.
  7. GATE 2023 ST Q44 (2 marks, Multiple choice) – For U⊂ℝⁿ let L(U)=span(U). For T,S⊂ℝⁿ,which statement is NOT true?
  8. GATE 2023 ST Q51 (2 marks, Multiple select) – A real 3×3 matrix satisfies A(0,1,1)ᵀ=(4,0,0)ᵀ,A(1,0,1)ᵀ=(0,4,0)ᵀ,and A(1,1,0)ᵀ=(0,0,4)ᵀ. Which hold?
  9. GATE 2024 ST Q13 (1 mark, Multiple choice) – For a real 3×3 A, which claim is not true?
  10. GATE 2024 ST Q14 (1 mark, Multiple choice) – Triples u₁,u₂,u₃ and v₁,v₂,v₃ lie in ℝ⁴ and span U,V respectively. I:if dim(U∩V)=2 and dim U=3, the v triple is dependent. II:if U+V=ℝ⁴, at least one…
  11. GATE 2024 ST Q16 (1 mark, Multiple choice) – A real 2×3 matrix has nonzero rows a,b and rank 1. Let W=ker A,W₁=x:a·x=0,W₂=x:b·x=0. I:W=W₁∩W₂. II:W₁=W₂. Which hold?
  12. GATE 2025 ST Q27 (1 mark, Numerical answer) – T:ℝ³→ℝ³ sends (x₁,x₂,x₃) to (3x₁+5x₂+x₃,x₃,2x₁+2x₃). Find rank T (integer).
  13. GATE 2026 ST Q13 (1 mark, Multiple choice) – In ℝ⁴ let W₁=x:x₁+x₂+x₃+x₄=0,W₂=x:x₁+2x₂+3x₃+4x₄=0. Find dim(W₁∩W₂).
  14. GATE 2026 ST Q25 (1 mark, Numerical answer) – The system x+y+z=1,2x+y+3z=6,3x+2y+kz=k+1 has no solution. Find k (integer).
  15. GATE 2026 ST Q39 (2 marks, Multiple choice) – Consider x+2y−z=a,x+y+3z=b,2x+3y+2z=c. I:a solution exists for every (a,b,c)∈ℝ³. II:for (a,b,c)=0 the solutions are (−7t,4t,t):t∈ℝ. Which hold?