GATE Engineering Mathematics: Linear Algebra – Previous Year Questions

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

1. GATE 2020 BM Q27 (1 mark, Numerical answer) – A nonsingular 3×3 matrix P has eigenvalues 1,2,3. Find trace(P⁻¹), to two decimals.
2. GATE 2021 BM Q36 (2 marks, Multiple choice) – A nonsingular 2×2 matrix A has trace 12 and determinant 32. Find the eigenvalues of A⁻¹.
3. GATE 2022 BM Q11 (1 mark, Multiple choice) – A=pmatrix3&-2\\4&-2pmatrix and I is the 2×2 identity. If A2=kA-2I, find k.
4. GATE 2023 BM Q13 (1 mark, Multiple choice) – For Q=pmatrix1&-2\\2&1pmatrix, which statement is true?
5. GATE 2023 BM Q14 (1 mark, Multiple choice) – For A=pmatrix1&1&0\\1&-1&0\\1&-1&1pmatrix, which vector is an eigenvector corresponding to eigenvalue 1?
6. GATE 2024 BM Q36 (2 marks, Multiple choice) – Find the eigenvalues of A=pmatrix1&-1\\2&-2pmatrix.
7. GATE 2025 BM Q17 (1 mark, Multiple choice) – A 3×3 matrix M has determinant 8. Multiply every element by −2. Find the new determinant.
8. GATE 2025 BM Q21 (1 mark, Multiple choice) – Find the eigenvalues of pmatrix&-\\&pmatrix.
9. GATE 2026 BM Q15 (1 mark, Multiple choice) – Find the eigenvalues of P=pmatrix1&-2\\3&-4pmatrix.
10. GATE 2026 BM Q18 (1 mark, Multiple choice) – Invertible n×n matrix P satisfies P³=I. Which must hold?
11. GATE 2026 BM Q58 (2 marks, Numerical answer) – Vector (2,a)T is an eigenvector for the smallest eigenvalue of pmatrix1&2\\2&1pmatrix. Find a as an integer.