The GATE Grind

GATE Engineering Mathematics: Linear Algebra – Previous Year Questions

30 GATE previous year questions on Linear Algebra (Engineering Mathematics, Computer Science) with answers and explanations, from every paper.

  1. GATE 2017 CS Q13 (1 mark, Multiple choice) – Let c 1, , c n be scalars, not all zero, such that i=1n c i a i = 0 where a i are column vectors in Rn. Consider the set of linear equations Ax = b…
  2. GATE 2017 CS Q40 (2 marks, Multiple choice) – Let u and v be two vectors in R2 whose Euclidean norms satisfy \ u\ = 2\ v\ . What is the value of such that w = u + v bisects the angle between u and…
  3. GATE 2017 CS Q41 (2 marks, Multiple choice) – Let A be n n real valued square symmetric matrix of rank 2 with i=1n j=1n A ij2 = 50. Consider the following statements. (I) One eigenvalue must be in…
  4. GATE 2016 CS Q15 (1 mark, Numerical answer) – Two eigenvalues of a 3 3 real matrix P are (2 + -1) and 3. The determinant of P is .
  5. GATE 2015 CS Q33 (1 mark, Numerical answer) – In the LU decomposition of the matrix bmatrix 2 & 2 \\ 4 & 9 bmatrix, if the diagonal elements of U are both 1, then the lower diagonal entry l 22 of…
  6. GATE 2015 CS Q59 (2 marks, Multiple choice) – Consider the following 2 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are -1 and 7. What are…
  7. GATE 2018 CS Q27 (1 mark, Numerical answer) – Consider a matrix A=uvT where u=pmatrix1\\2pmatrix, v=pmatrix1\\1pmatrix. Note that vT denotes the transpose of v. The largest eigenvalue of A is .
  8. GATE 2018 CS Q36 (2 marks, Multiple choice) – Consider a matrix P whose only eigenvectors are the multiples of bmatrix1\\4bmatrix. Consider the following statements. (I) P does not have an inverse…
  9. GATE 2019 CS Q19 (1 mark, Multiple choice) – Let X be a square matrix. Consider the following two statements on X. I. X is invertible. II. Determinant of X is non-zero. Which one of the following…
  10. GATE 2019 CS Q54 (2 marks, Numerical answer) – Consider the following matrix: R=bmatrix1&2&4&8\\1&3&9&27\\1&4&16&64\\1&5&25&125bmatrix The absolute value of the product of Eigen values of R is .
  11. GATE 2026 CS (CS2) Q31 (1 mark, Numerical answer) – Consider the system of linear equations ax+y=b 16x+ay=24 Suppose the values of a and b are chosen such that the system produces multiple solutions.…
  12. GATE 2026 CS (CS2) Q62 (2 marks, Numerical answer) – The determinant of a 44 matrix A is 3. The value of the determinant of 2A is . (answer in integer)
  13. GATE 2026 CS (CS1) Q12 (1 mark, Multiple choice) – Consider 4 4 matrices with their elements from \0, 1\. The number of such matrices with even number of 1s in every row and every column is
  14. GATE 2026 CS (CS1) Q13 (1 mark, Multiple choice) – For n > 1, the maximum multiplicity of any eigenvalue of an n n matrix with elements from R is
  15. GATE 2026 CS (CS1) Q20 (1 mark, Multiple select) – Let n > 1. Consider an n n matrix M with its elements from R. Let the vector (0, 1, 0, 0, , 0) Rn be in the null space of M. Which of the following…
  16. GATE 2025 CS (CS2) Q11 (1 mark, Multiple choice) – If A = pmatrix1 & 2\\2 & -1pmatrix, then which ONE of the following is A8 ?
  17. GATE 2025 CS (CS2) Q14 (1 mark, Multiple choice) – Let L, M, and N be non-singular matrices of order 3 satisfying the equations L2 = L-1, M = L8 and N = L2. Which ONE of the following is the value of…
  18. GATE 2025 CS (CS2) Q44 (2 marks, Multiple select) – Consider a system of linear equations PX = Q where P R33 and Q R31. Suppose P has an LU decomposition, P = LU, where L = bmatrix1 & 0 & 0\ 21 & 1 & 0\…
  19. GATE 2025 CS (CS1) Q23 (1 mark, Multiple select) – Consider the given system of linear equations for variables x and y, where k is a real-valued constant. Which of the following option(s) is/are…
  20. GATE 2025 CS (CS1) Q41 (2 marks, Multiple choice) – Let A be a 2 2 matrix as given. A = bmatrix 1 & 1 \\ 1 & -1 bmatrix What are the eigenvalues of the matrix A13 ?
  21. GATE 2024 CS (CS2) Q47 (2 marks, Multiple select) – Let A be an n n matrix over the set of all real numbers R. Let B be a matrix obtained from A by swapping two rows. Which of the following statements…
  22. GATE 2024 CS (CS1) Q12 (1 mark, Multiple choice) – The product of all eigenvalues of the matrix bmatrix1&2&3\\4&5&6\\7&8&9bmatrix is
  23. GATE 2024 CS (CS1) Q49 (2 marks, Multiple select) – Let A be any n m matrix, where m>n. Which of the following statements is/are TRUE about the system of linear equations Ax=0?
  24. GATE 2023 CS Q18 (1 mark, Multiple choice) – Let A=bmatrix1&2&3&4\\4&1&2&3\\3&4&1&2\\2&3&4&1bmatrix and B=bmatrix3&4&1&2\\4&1&2&3\\1&2&3&4\\2&3&4&1bmatrix. Let det(A) and det(B) denote the…
  25. GATE 2023 CS Q30 (1 mark, Numerical answer) – Let A be the adjacency matrix of the graph with vertices 1, 2, 3, 4, 5. [Figure: a graph on vertices 1-5.] Let 1, 2, 3, 4, 5 be the five eigenvalues…
  26. GATE 2022 CS Q20 (1 mark, Multiple choice) – Consider the following two statements with respect to the matrices A m n, B n m, C n n and D n n. Statement 1: tr(AB) = tr(BA) Statement 2: tr(CD) =…
  27. GATE 2022 CS Q45 (2 marks, Multiple choice) – Consider solving the following system of simultaneous equations using LU decomposition. x 1+x 2-2x 3=4 x 1+3x 2-x 3=7 2x 1+x 2-5x 3=7 where L and U…
  28. GATE 2022 CS Q53 (2 marks, Multiple select) – Which of the following is/are the eigenvector(s) for the matrix given below? pmatrix-9&-6&-2&-4\\-8&-6&-3&-1\\20&15&8&5\\32&21&7&12pmatrix
  29. GATE 2021 CS Q62 (2 marks, Numerical answer) – Consider the following matrix. pmatrix 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 pmatrix The largest eigenvalue of the above matrix is .
  30. GATE 2020 CS Q37 (2 marks, Multiple choice) – Let A and B be two n n matrices over real numbers. Let rank(M) and (M) denote the rank and determinant of a matrix M, respectively. Consider the…