GATE Linear Algebra: Eigenvalues, eigenvectors and determinant – Previous Year Questions

7 GATE previous year questions on Eigenvalues, eigenvectors and determinant (Linear Algebra, Data Science and Artificial Intelligence) with answers and explanations, from every paper.

1. GATE 2024 DA Q13 (1 mark, Multiple choice) – Consider the matrix M = bmatrix 2 & -1 \\ 3 & 1 bmatrix. Which ONE of the following statements is TRUE?
2. GATE 2024 DA Q35 (1 mark, Numerical answer) – Consider the 3 3 matrix M = bmatrix 1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6 bmatrix. The determinant of (M2 + 12M) is .
3. GATE 2025 DA Q13 (1 mark, Multiple choice) – The sum of the elements in each row of A Rn n is 1. If B = A3 - 2A2 + A, which one of the following statements is correct (for x Rn)?
4. GATE 2025 DA Q28 (1 mark, Multiple select) – Let A = I n + xx, where I n is the n n identity matrix and x Rn, x x = 1. Which of the following options is/are correct?
5. GATE 2026 DA Q21 (1 mark, Multiple choice) – Let M = pmatrix & \\ - & pmatrix be a 2 2 matrix, where = 25, and I 2 = pmatrix 1 & 0 \\ 0 & 1 pmatrix. Which of the following options is equal to…
6. GATE 2026 DA Q46 (2 marks, Multiple choice) – Let 1, 2, 3 be the eigenvalues of the matrix bmatrix 1 & 0 & 0 \\ 0 & t & t \\ 0 & - t & t bmatrix, where t [-, ] is in radians. Which one of the…
7. GATE 2026 DA Q65 (2 marks, Numerical answer) – Let A = (I n - 1n11T) be a matrix, where 1 = (1, 1, 1, , 1)T Rn and I n is the identity matrix of order n. The value of x S xTAx, where S = \x Rn xTx…