What are common examiner traps in Linear Algebra?

Applying det(cA) = c det(A) instead of cn det(A) for an n × n matrix. Assuming det(A+B) = det(A) + det(B) or rank(AB) = rank(A)rank(B), which are false in general. Forgetting that non-zero eigenvalues in outer product uvT are given by the dot product vTu, mistakenly thinking rank equals n. Confusing algebraic multiplicity (root multiplicity in characteristic polynomial) with geometric multiplicity (number of linearly independent eigenvectors). Overlooking that complex eigenvalues of real matrices always occur in conjugate pairs (a ± ib). Missing that diagonal entries in Doolittle LU decomposition have Lii = 1, whereas in Crout Uii = 1. Sign errors when substituting back relationships between variables (e.g., confusing x = -2y = -4z with x = +4z). Forgetting that eigenvectors must be non-zero vectors (i.e. ≠ 0). Failing to reduce the system properly when one row is all zeros. Sign errors when computing root values (e.g., a2 = 16 a = ± 4), though in this case both roots yield the same product a · b. Confusing the condition for no solution ((a1)/(a2) = (b1)/(b2) ≠ (c1)/(c2)) with the condition for infinitely many solutions. Forgetting that 'multiple solutions' in a linear system implies infinitely many solutions. Assuming trace is invariant under row swaps (swapping rows alters diagonal entries unless off-diagonal elements match). Assuming structural properties like symmetry are preserved after applying elementary row operations without corresponding column operations. Assuming underdetermined systems (m n) never have solutions, forgetting consistent/redundant systems or homogeneous systems (Ax = 0). Failing to distinguish between 'all systems have a solution' versus 'there exists a system that has a solution'.

GATE CSE Linear Algebra PYQs

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Linear Algebra in GATE CS is consistently tested across fundamental properties of matrices, systems of linear equations, and spectral theory (eigenvalues and ei… Read the guide

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Questions: 46
Asked: since 2011
Total marks: 64
Marks / paper: 2.5

46 GATE CSE Linear Algebra PYQs with solutions, from Engineering Mathematics. Browse all GATE CSE PYQs.

Questions

Every official paper that asked this topic, newest first.

What GATE tests in Linear Algebra

Concepts, patterns, formulas and traps distilled from past papers.

Linear Algebra in GATE CS is consistently tested across fundamental properties of matrices, systems of linear equations, and spectral theory (eigenvalues and eigenvectors). Questions heavily leverage algebraic identities such as trace-sum, determinant-product relations, rank-nullity theorem, and structural properties like orthogonality of symmetric matrices and LU decomposition. Problems frequently appear as 1-mark and 2-mark conceptual MCQs/MSQs or direct numerical computation NATs.

Core concepts

Characteristic equation and algebraic/geometric multiplicity of eigenvalues
Eigenvalues of special matrices (triangular, symmetric, skew-symmetric, orthogonal, circulant, rank-1 outer product)
Orthogonality of eigenvectors for distinct eigenvalues of real symmetric matrices
Rank, Nullity, and fundamental subspaces of matrices
Consistency criteria for linear systems Ax = b via row-echelon form and ranks
Properties of determinants (multiplicative det(AB) = det(A)det(B), scaling det(cA) = cn det(A), row swap sign inversion)
LU Decomposition and solving systems via LY = b and UX = Y
Vector space dimensions, linear independence of orthogonal sets, and angle bisectors
Definition of Eigenvectors and Eigenvalues
Solving homogeneous linear systems (A - I)X = 0
Parametric representation of eigenspaces ( ≠ 0, ∈ R)
Row echelon reduction of (A - I)
System of linear equations consistency
Infinite (multiple) solutions condition
Matrix rank and augmented matrix rank
Proportionality of linear constraints
Elementary row operations and their effect on determinants
Matrix invertibility and non-singularity
Matrix symmetry (A = AT)
Trace of a matrix and change of diagonal elements under permutation
Underdetermined systems (m < n) and their potential inconsistency
Overdetermined systems (m > n) and their consistency conditions
Square systems (m = n) and non-singular existence
Homogeneous vs non-homogeneous system solution spaces
Universal vs existential quantification over linear systems

Recurring question patterns

Eigenvalue Calculation via Trace and Determinant
CommonMCQ1 marks2025, 2024, 2016, 2015
Given a matrix with unknown parameters or asking for powers $A^k$ , candidates compute eigenvalues using the characteristic polynomial or sum/product relations (trace and determinant).
Rank-Nullity and System of Linear Equations Consistency
Commonmixed2 marks2026, 2025, 2024, 2021
Determining the number of solutions (unique, infinite, no solution) or dimension of the null space / rank of coefficient and augmented matrices for $Ax = b$ or $Ax = 0$ .
Outer Product Matrix Properties
OccasionalNAT1 marks2018, 2014
Matrices formed by $A = u v^T$ where rank is 1, yielding $n-1$ zero eigenvalues and one eigenvalue equal to the trace ( $v^T u$ ), with $\det(A) = 0$ for $n > 1$ .
LU Decomposition Properties and Forward/Backward Substitution
Occasionalmixed2 marks2025, 2022, 2015
Factorization $A = LU$ (Doolittle or Crout forms) to find specific matrix entries $l_{ij}, u_{ij}$ , or checking theoretical invertibility/singularity properties.
Determinant Properties and Scalar Scaling
CommonNAT1 marks2026, 2025, 2023, 2016
Evaluating $\det(cA) = c^n \det(A)$ , row swap sign changes, or complex conjugate roots in real matrices.
Subspace Dimensions and Intersection
RareNAT1 marks2014
Applying Grassmann's dimension formula $\dim(V_1 \cap V_2) = \dim(V_1) + \dim(V_2) - \dim(V_1 + V_2)$ under dimension constraints.
Eigenvector Determination for a Known Eigenvalue
CommonMCQ1 marks2015
Given a $3 \times 3$ matrix and a specific eigenvalue $\lambda$ , solve the homogeneous system $(A - \lambda I)X = 0$ to identify the parameterized set of non-zero eigenvectors.
Parameter Conditions for Infinite Solutions in Linear Systems
CommonNAT1 marks2026
Given a parameterized 2x2 or 3x3 non-homogeneous system of linear equations, candidates determine values or products of parameters such that the system has infinitely many (multiple) solutions.
Effect of Elementary Row Operations on Matrix Properties
CommonMSQ2 marks2024
Given a matrix $A$ and a transformed matrix $B$ obtained via an elementary operation (e.g., swapping two rows), candidates evaluate which properties (determinant, invertibility, symmetry, trace, rank) are preserved or modified.
Dimension and Consistency Statement Analysis
RareMCQ1 marks2016
Evaluating the truth value of propositions regarding whether systems with $m < n$ , $m > n$ , or $m = n$ are guaranteed to have solutions, no solutions, or at least one consistent instance.

Formulas GATE uses

Trace-Eigenvalue Sum Identity

\text{Trace}(A) = \sum_{i=1}^{n} a_{ii} = \sum_{i=1}^{n} \lambda_i

Used to find missing eigenvalues, determine parameter values, or sum all eigenvalues of adjacency and general matrices.

Determinant-Eigenvalue Product Identity

\det(A) = \prod_{i=1}^{n} \lambda_i

Used to find the product of eigenvalues, determinant of inverses, or verify singularity.

Scalar Determinant Property

\det(cA) = c^n \det(A)

Used when scaling an $n \times n$ matrix $A$ by a constant scalar $c$ .

Rank-Nullity Theorem

\text{Rank}(A) + \text{Nullity}(A) = n

Used for an $m \times n$ matrix to find the dimension of the solution space of homogeneous systems $Ax = 0$ .

Subspace Dimension Theorem

\dim(V_1 + V_2) = \dim(V_1) + \dim(V_2) - \dim(V_1 \cap V_2)

Used to determine minimal or maximal possible dimensions of intersecting subspaces.

Matrix Power Eigenvalues

A v = \lambda v \implies A^k v = \lambda^k v

Used to find eigenvalues of higher matrix powers like $A^{13}$ or $A^{19}$ .

Outer Product Matrix Eigenvalues

A = u v^T \implies \lambda = v^T u \text{ (with multiplicity 1), and } \lambda = 0 \text{ (multiplicity } n-1)

Used when a matrix is presented as the outer product of two column vectors.

Frobenius Norm and Eigenvalue Identity for Symmetric Matrices

\sum_{i=1}^n \sum_{j=1}^n A_{ij}^2 = \text{Tr}(A^T A) = \sum_{k=1}^n \lambda_k^2

Used to bound eigenvalues of real symmetric matrices given the sum of squares of entries.

Eigenvector System Equation

(A - \lambda I)X = 0

Used to find the non-trivial solution vector $X \neq 0$ corresponding to a known eigenvalue $\lambda$ of matrix $A$ .

Condition for Infinitely Many Solutions in a 2-Variable System

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

Used when a 2-equation system $a_1 x + b_1 y = c_1$ and $a_2 x + b_2 y = c_2$ is specified to have multiple (infinite) solutions.

Rouché-Capelli Theorem for Infinite Solutions

\text{rank}(A) = \text{rank}([A|B]) < n

General matrix condition for a system $Ax = B$ of $n$ variables to be consistent with infinitely many solutions.

Determinant under Row Interchange

\det(B) = -\det(A)

Used when finding the determinant of a matrix $B$ obtained by swapping two rows of matrix $A$ .

Invertibility Condition via Determinant

A \text{ is invertible} \iff \det(A) \neq 0

Used to relate the invertibility of transformed matrix $B$ to the invertibility of $A$ .

Matrix Trace Definition

\operatorname{trace}(A) = \sum_{i=1}^{n} a_{ii}

Used to verify whether trace is invariant under row operations.

Consistency Condition (Rouché-Capelli Theorem)

\text{rank}(A) = \text{rank}([A \mid b])

Determining whether a non-homogeneous system $Ax = b$ has at least one solution.

Homogeneous System Trivial Solution

Ax = 0 \implies x = 0 \text{ is always a solution}

Disproving claims that overdetermined systems ( $m > n$ ) can never have solutions.

Examiner traps in Linear Algebra

Applying det(cA) = c det(A) instead of cn det(A) for an n × n matrix.
Assuming det(A+B) = det(A) + det(B) or rank(AB) = rank(A)rank(B), which are false in general.
Forgetting that non-zero eigenvalues in outer product uvT are given by the dot product vTu, mistakenly thinking rank equals n.
Confusing algebraic multiplicity (root multiplicity in characteristic polynomial) with geometric multiplicity (number of linearly independent eigenvectors).
Overlooking that complex eigenvalues of real matrices always occur in conjugate pairs (a ± ib).
Missing that diagonal entries in Doolittle LU decomposition have Lii = 1, whereas in Crout Uii = 1.
Sign errors when substituting back relationships between variables (e.g., confusing x = -2y = -4z with x = +4z).
Forgetting that eigenvectors must be non-zero vectors (i.e. ≠ 0).
Failing to reduce the system properly when one row is all zeros.
Sign errors when computing root values (e.g., a2 = 16 a = ± 4), though in this case both roots yield the same product a · b.
Confusing the condition for no solution ((a1)/(a2) = (b1)/(b2) ≠ (c1)/(c2)) with the condition for infinitely many solutions.
Forgetting that 'multiple solutions' in a linear system implies infinitely many solutions.
Assuming trace is invariant under row swaps (swapping rows alters diagonal entries unless off-diagonal elements match).
Assuming structural properties like symmetry are preserved after applying elementary row operations without corresponding column operations.
Assuming underdetermined systems (m < n) always have infinite solutions, forgetting they can be inconsistent (e.g., x+y=1 and x+y=2).
Assuming overdetermined systems (m > n) never have solutions, forgetting consistent/redundant systems or homogeneous systems (Ax = 0).
Failing to distinguish between 'all systems have a solution' versus 'there exists a system that has a solution'.

How questions are asked

Direct eigenvalue/determinant evaluation for numerical answers (NAT)
Multi-statement correctness evaluation on matrix rank/nullity/diagonalizability (MCQ/MSQ)
Parametric system solvability classification (infinite vs no solution conditions)
Step-by-step LU decomposition entry identification
Outer product or pattern-based rank and determinant computation
Parametrized set identification for eigenvectors
Find parameter values for infinite/unique/no solutions in linear equations (NAT)
Multi-select property verification under matrix transformation
Multi-statement truth verification (Roman numeral options I, II, III)

Paper trends

Transition from direct 1-mark eigenvalue/determinant computations (2011-2016) toward multi-statement theoretical MSQs analyzing subspace properties, nullity, and LU conditions (2021-2026).
2026, 2025, 2024, 2022, 2021, 2016, 2011
Integration of graph theory representations (adjacency matrix diagonal self-loops yielding trace and sum of eigenvalues).
2023
Consistent presence of parameter-dependent linear systems ( $Ax = b$ containing parameter $k$ asking for no-solution/infinite-solution ranges).
2025, 2015
Questions directly provide the eigenvalue to focus the evaluation on finding the null space basis / eigenspace parametrization rather than solving the cubic characteristic polynomial.
2015
Direct testing of linear equation consistency conditions using algebraic parameter constraints in NAT format for 1 mark.
2026
Introduction of conceptual Multiple Select Questions (MSQ) testing fundamental properties (determinants, invertibility, symmetry, trace) under matrix transformations rather than purely numerical computation.
2024
Focus on pure conceptual counterexamples and quantifier precision rather than numerical row reduction.
2016

Difficulty notes

Easy questions test direct property recall such as diagonal matrix eigenvalues, $\det(cA)$ , trace sum, or single-step $2 \times 2$ characteristic roots. Medium questions require combining multi-step concepts, such as LU factorization multipliers, parameter-dependent systems of 3+ equations, or verifying eigenvectors by matrix-vector multiplication. Hard questions demand deep theoretical reasoning involving quadratic matrix forms, Frobenius norms related to eigenvalue bounds, or abstract subspace dimension inequalities.