Practice tests · GATE

GATE linear algebra: 5 practice questions with solutions

By the PrepMatic team · Updated 9 Oct 2026 · 4 min read

Linear algebra is part of the engineering-mathematics block in the engineering papers, and a full subject section in Data Science and AI (DA). The questions are short, the formulas are few, and a one-line property often replaces a page of calculation. The five original questions below test consistency of systems, eigenvalues, determinants, eigenvectors and LU decomposition.

What GATE tests in linear algebra

Across the 2027 syllabi the core list is matrices and determinants, linear systems, rank, eigenvalues and eigenvectors, and LU decomposition (which CS names explicitly). DA adds quadratic forms, projection matrices and singular value decomposition (SVD). EC, EE, ME, CE and RA all keep linear algebra in their maths section. The wording changed in the 2027 revision, so check your own paper's list on the official syllabus page. The questions fall into a few patterns:

• Rank and consistency of Ax = b (unique, infinite or no solution).
• Eigenvalue shortcuts: sum equals trace, product equals determinant, eigenvalues of a polynomial in A.
• Determinant properties, including scalar multiples, inverses and the adjoint.
• Eigenvector calculation for 2 × 2 and triangular matrices.
• Factorisations, such as finding an entry of L or U.

The real paper also has MSQ and NAT questions; all five below are MCQ.

Practice questions

Question 1

Consider the system x + y + z = 1, x + 2y + 3z = 4, 2x + 3y + 4z = 6. The system has:

• A. a unique solution
• B. no solution
• C. infinitely many solutions
• D. exactly two solutions

Question 2

Let A = [[4, 1], [2, 3]], written row by row. The trace of the matrix A² − 3A + 2I is:

• A. 6
• B. 8
• C. 10
• D. 12

Question 3

A is a 3 × 3 real matrix with det(A) = 4. The value of det(adj(2A)) is:

• A. 256
• B. 512
• C. 1024
• D. 4096

Question 4

An eigenvector of the matrix [[3, 1], [0, 2]] corresponding to the eigenvalue 2 is:

• A. (1, −1)
• B. (1, 0)
• C. (1, 1)
• D. (0, 1)

Question 5

The matrix A = [[2, 1, 1], [4, 3, 3], [8, 7, 9]] is factorised as A = LU, where L is lower triangular with unit diagonal and U is upper triangular. The (3, 3) entry of U is:

• A. −2
• B. 0
• C. 1
• D. 2

Answers and explanations

1. B. Add the first two equations: 2x + 3y + 4z = 5, but the third equation says it equals 6. The coefficient matrix has rank 2 (row 3 = row 1 + row 2), while the augmented matrix has rank 3, so the system is inconsistent. A linear system never has "exactly two" solutions.
2. D. The trace of a matrix equals the sum of its eigenvalues, and f(A) has eigenvalues f(λ). Here trace(A) = 7 and det(A) = 10, so λ² − 7λ + 10 = 0 gives λ = 5 and 2. For f(λ) = λ² − 3λ + 2 the values are 12 and 0, and their sum is 12. Direct check: A² = [[18, 7], [14, 11]], trace 29; 29 − 21 + 4 = 12.
3. C. For an n × n matrix M, det(adj M) = det(M)^(n−1). Here M = 2A, and det(2A) = 2³ × 4 = 32. So det(adj(2A)) = 32² = 1024. The usual slip is to use the exponent n instead of n − 1.
4. A. Solve (A − 2I)v = 0. A − 2I = [[1, 1], [0, 0]], so x + y = 0 and v = (1, −1) up to scaling. Option B, (1, 0), is the eigenvector for eigenvalue 3.
5. D. Eliminate below the first pivot: row 2 − 2 × row 1 and row 3 − 4 × row 1 give multipliers 2 and 4. The new rows are (0, 1, 1) and (0, 3, 5). Next, row 3 − 3 × row 2 gives (0, 0, 2), with multiplier 3. So U = [[2, 1, 1], [0, 1, 1], [0, 0, 2]] and L = [[1, 0, 0], [2, 1, 0], [4, 3, 1]]. As a check, det(A) = 2 × 1 × 2 = 4.

How to prepare linear algebra

• Learn the four standard facts first: trace is the sum of eigenvalues, determinant is their product, a triangular matrix has its diagonal as eigenvalues, and f(A) has eigenvalues f(λ). They answer many questions with no matrix arithmetic at all.
• Master row reduction. Rank, consistency, null space and LU all come from the same elimination. Practise until you can reduce a 3 × 3 augmented matrix in under two minutes without errors.
• Memorise the determinant rules: det(kA) = kⁿ det(A), det(AB) = det(A) det(B), det(A⁻¹) = 1/det(A), det(Aᵀ) = det(A), det(adj A) = det(A)ⁿ⁻¹.
• For DA, add the extras: orthogonal projection matrices and their properties, quadratic forms and definiteness, and what SVD gives you (rank, singular values, low-rank approximation).
• Solve by hand first, then check with the properties. If your answer breaks the trace or determinant rule, you have found an arithmetic slip.

The GATE engineering mathematics guide shows how linear algebra fits with calculus, probability and differential equations for your branch; continue with calculus and differential equations. DA aspirants will find the paper layout in the GATE Data Science and AI guide.

Take the full topic test on PrepMatic

These five questions are a sample. PrepMatic's topic tests run about 15 questions in 30 minutes, and the free assessment shows your weakest subjects, so you can decide whether linear algebra needs your next week.

← All practice tests