What are common examiner traps in Linear Algebra?

Applying (cA) = c (A) instead of cn (A) for an n n matrix. Assuming (A+B) = (A) + (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 ( a1a2 = b1b2 c1c2) 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'.

Linear Algebra

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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 eigenvectors). Questio… Guide

Type

GATE CSE 2026 Set 1Q13MCQ1 markEasyFor $n > 1$ , the maximum multiplicity of any eigenvalue of an $n \times n$ matrix with elements from $\mathbb{R}$ is
GATE CSE 2026 Set 1Q20MSQ1 markEasyLet $n > 1$ . Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$ . Let the vector $(0, 1, 0, 0, \dots, 0) \in \mathbb{R}^n$ be in the null space of $M$ . Which of the following options is/are always correct?
GATE CSE 2026 Set 2Q31NAT1 markEasyConsider the system of linear equations given below. $ax + y = b$ $16x + ay = 24$ Suppose the values of $a$ and $b$ are chosen such that the system of linear equations produce multiple solutions. Then the product of $a$ and $b$ is _______. (answer in integer)
GATE CSE 2026 Set 2Q62NAT2 marksEasyThe determinant of a $4 \times 4$ matrix $A$ is $3$ . The value of the determinant of $2A$ is __________. (answer in integer)
GATE CSE 2025 Set 1Q23MSQ1 markEasyConsider 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 CORRECT? $x + ky = 1$ $kx + y = -1$
GATE CSE 2025 Set 1Q41MCQ2 marksEasyLet $A$ be a $2 \times 2$ matrix as given. $A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$ What are the eigenvalues of the matrix $A^{13}$ ?
GATE CSE 2025 Set 2Q11MCQ1 markEasyIf $A = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix}$ , then which ONE of the following is $A^8$ ?
GATE CSE 2025 Set 2Q14MCQ1 markEasyLet $L, M$ , and $N$ be non-singular matrices of order 3 satisfying the equations $L^2 = L^{-1},\ M = L^8 \text{ and } N = L^2.$ Which ONE of the following is the value of the determinant of $(M - N)$ ?
GATE CSE 2025 Set 2Q44MSQ2 marksMediumConsider a system of linear equations $PX = Q$ where $P \in \mathbb{R}^{3\times 3}$ and $Q \in \mathbb{R}^{3\times 1}$ . Suppose $P$ has an LU decomposition, $P = LU$ , where $L = \begin{bmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{bmatrix} \text{ and } U = \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix}.$ …
GATE CSE 2024 Set 1Q12MCQ1 markEasyThe product of all eigenvalues of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ is
GATE CSE 2024 Set 1Q49MSQ2 marksMediumLet $A$ be any $n \times m$ matrix, where $m > n$ . Which of the following statements is/are TRUE about the system of linear equations $Ax = \mathbf{0}$ ?
GATE CSE 2024 Set 2Q47MSQ2 marksEasyLet $A$ be an $n \times n$ matrix over the set of all real numbers $\mathbb{R}$ . Let $B$ be a matrix obtained from $A$ by swapping two rows. Which of the following statements is/are TRUE?
GATE CSE 2023Q18MCQ1 markEasyLet $A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{bmatrix}.$ Let $\det(A)$ and $\det(B)$ denote the determinants of the matrices $A$ and $B$ , respectively.…
GATE CSE 2023Q30NAT1 markEasyDiagramLet $A$ be the adjacency matrix of the graph with vertices $\{1, 2, 3, 4, 5\}$ . Let $\lambda_1, \lambda_2, \lambda_3, \lambda_4,$ and $\lambda_5$ be the five eigenvalues of $A$ . Note that these eigenvalues need not be distinct. The value of $\lambda_1 + \lambda_2 + \lambda_3 + \lambda_4 + \lambda_5 =$ …
GATE CSE 2022Q20MCQ1 markEasyConsider the following two statements with respect to the matrices $A_{m \times n}$ , $B_{n \times m}$ , $C_{n \times n}$ and $D_{n \times n}$ . Statement 1: $tr(AB) = tr(BA)$ Statement 2: $tr(CD) = tr(DC)$ where $tr()$ represents the trace of a matrix. Which one of the following holds?
GATE CSE 2022Q45MCQ2 marksMediumConsider solving the following system of simultaneous equations using LU decomposition. $\begin{aligned} x_1 + x_2 - 2x_3 &= 4 x_1 + 3x_2 - x_3 &= 7 \\ 2x_1 + x_2 - 5x_3 &= 7 \end{aligned}$ where $L$ and $U$ are denoted as $L = \begin{pmatrix} L_{11} & 0 & 0 \\ L_{21} & L_{22} & 0 \\ L_{31} & L_{32} & L_{33} \end{pmatrix}, \quad U = \begin{pmatrix} U_{11} & U_{12} & U_{13} \\ 0 & U_{22} & U_{23} \\ 0 & 0 & U_{33} \end{pmatrix}$ …
GATE CSE 2022Q53MSQ2 marksMediumWhich of the following is/are the eigenvector(s) for the matrix given below? $\begin{pmatrix} -9 & -6 & -2 & -4 \\ -8 & -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32 & 21 & 7 & 12 \end{pmatrix}$
GATE CSE 2021 Set 1Q52NAT2 marksEasyConsider the following matrix. $\begin{pmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{pmatrix}$ The largest eigenvalue of the above matrix is _____.
GATE CSE 2021 Set 2Q24NAT1 markEasySuppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $Px = 0$ is a scalar multiple of $[2\; 5\; 4\; 3\; 1]^T$ . The rank of $P$ is _______.
GATE CSE 2021 Set 2Q37MCQ2 marksEasyFor two $n$ -dimensional real vectors $P$ and $Q$ , the operation $s(P, Q)$ is defined as follows: $s(P, Q) = \sum_{i=1}^n (P[i] \cdot Q[i])$ Let $\mathcal{L}$ be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors $P, Q \in \mathcal{L}$ , $s(P, Q) = 0$ . What is the…
GATE CSE 2020Q27MCQ2 marksEasyLet $A$ and $B$ be two $n \times n$ matrices over real numbers. Let $\text{rank}(M)$ and $\det(M)$ denote the rank and determinant of a matrix $M$ , respectively. Consider the following statements. I. \( $\text{rank}$ (AB) = $\text{rank}$
GATE CSE 2019Q9MCQ1 markEasyLet $X$ 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 is TRUE?
GATE CSE 2018Q17NAT1 markEasyConsider a matrix $A = uv^T$ where $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ , $v = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ . Note that $v^T$ denotes the transpose of $v$ . The largest eigenvalue of $A$ is ______.
GATE CSE 2018Q26MCQ2 marksMediumConsider a matrix $P$ whose only eigenvectors are the multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$ . Consider the following statements. (I) $P$ does not have an inverse (II) $P$ has a repeated eigenvalue (III) $P$ cannot be diagonalized Which one of the following options is correct?
GATE CSE 2017 Set 1Q3MCQ1 markMediumLet $c_1, \ldots, c_n$ be scalars, not all zero, such that $\sum_{i=1}^n c_i a_i = 0$ where $a_i$ are column vectors in $\mathbf{R}^n$ . Consider the set of linear equations $Ax = b$ where $A = [a_1, \ldots, a_n]$ and $b = \sum_{i=1}^n a_i$ . The set of equations has
GATE CSE 2017 Set 1Q30MCQ2 marksMediumLet $u$ and $v$ be two vectors in $\mathbf{R}^2$ whose Euclidean norms satisfy $\|u\| = 2\|v\|$ . What is the value of $\alpha$ such that $w = u + \alpha v$ bisects the angle between $u$ and $v$ ?
GATE CSE 2017 Set 1Q31MCQ2 marksHardLet $A$ be $n \times n$ real valued square symmetric matrix of rank 2 with $\sum_{i=1}^n \sum_{j=1}^n A_{ij}^2 = 50$ . Consider the following statements. (I) One eigenvalue must be in $[-5, 5]$ (II) The eigenvalue with the largest magnitude must be strictly greater than 5 Which of the above statements about…
GATE CSE 2017 Set 2Q22NAT1 markEasyLet $P = \begin{bmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{bmatrix}$ and $Q = \begin{bmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{bmatrix}$ be two matrices. Then the rank of $P + Q$ is _________ .
GATE CSE 2017 Set 2Q52NAT2 marksMediumIf the characteristic polynomial of a $3 \times 3$ matrix $M$ over $\mathbb{R}$ (the set of real numbers) is $\lambda^3 - 4\lambda^2 + a\lambda + 30$ , $a \in \mathbb{R}$ , and one eigenvalue of $M$ is 2, then the largest among the absolute values of the eigenvalues of $M$ is _________.
GATE CSE 2016 Set 1Q5NAT1 markEasyTwo eigenvalues of a $3 \times 3$ real matrix $P$ are $(2 + \sqrt{-1})$ and $3$ . The determinant of $P$ is ________ .
GATE CSE 2016 Set 2Q4MCQ1 markMediumConsider the systems, each consisting of $m$ linear equations in $n$ variables. I. If $m < n$ , then all such systems have a solution II. If $m > n$ , then none of these systems has a solution III. If $m = n$ , then there exists a system which has a solution Which one of the following is CORRECT?
GATE CSE 2016 Set 2Q6NAT1 markEasySuppose that the eigenvalues of matrix $A$ are 1, 2, 4. The determinant of $(A^{-1})^T$ is __________.
GATE CSE 2015 Set 1Q18NAT1 markMediumIn the LU decomposition of the matrix $\begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}$ , if the diagonal elements of $U$ are both 1, then the lower diagonal entry $l_{22}$ of $L$ is _______.
GATE CSE 2015 Set 1Q36MCQ2 marksEasyConsider the following $2 \times 2$ 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 the values of $a$ and $b$ ? $A = \begin{pmatrix} 1 & 4 \\ b & a \end{pmatrix}$
GATE CSE 2015 Set 2Q5NAT1 markEasyThe larger of the two eigenvalues of the matrix $\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ is ______.
GATE CSE 2015 Set 3Q15MCQ1 markMediumIn the given matrix $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 1 & 0 \\ 1 & 2 & 1 \end{bmatrix}$ , one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are
GATE CSE 2015 Set 3Q33MCQ2 marksMediumIf the following system has non-trivial solution, $px + qy + rz = 0$ $qx + ry + pz = 0$ $rx + py + qz = 0,$ then which one of the following options is TRUE?
GATE CSE 2014 Set 1Q4NAT1 markMediumConsider the following system of equations: $\begin{aligned} 3x + 2y &= 1 \\ 4x + 7z &= 1 x + y + z &= 3 x - 2y + 7z &= 0 \end{aligned}$ The number of solutions for this system is ________
GATE CSE 2014 Set 1Q5NAT1 markEasyThe value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is ________ .
GATE CSE 2014 Set 2Q4NAT1 markEasyIf the matrix $A$ is such that $A = \begin{bmatrix} 2 \\ -4 \\ 7 \end{bmatrix} \begin{bmatrix} 1 & 9 & 5 \end{bmatrix}$ then the determinant of $A$ is equal to _________.
GATE CSE 2014 Set 2Q47NAT2 marksMediumThe product of the non-zero eigenvalues of the matrix $\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$ is _________.
GATE CSE 2014 Set 3Q4MCQ1 markEasyWhich one of the following statements is $\textbf{TRUE}$ about every $n \times n$ matrix with only real eigenvalues?
GATE CSE 2014 Set 3Q5NAT1 markEasyIf $V_1$ and $V_2$ are 4-dimensional subspaces of a 6-dimensional vector space $V$ , then the smallest possible dimension of $V_1 \cap V_2$ is _____.
GATE CSE 2013Q3MCQ1 markMediumWhich one of the following does NOT equal $\begin{vmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{vmatrix}$ ?
GATE CSE 2012Q11MCQ1 markMediumLet A be the $2 \times 2$ matrix with elements $a_{11} = a_{12} = a_{21} = +1$ and $a_{22} = -1$ . Then the eigenvalues of the matrix $A^{19}$ are
GATE CSE 2011Q40MCQ2 marksEasyConsider the matrix as given below. $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 0 & 0 & 3 \end{bmatrix}$ Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?

Topic guide

What does GATE test in Linear Algebra?

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 (AB) = (A) (B), scaling (cA) = cn (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
common · MCQ · 1 marks · 2025, 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
common · mixed · 2 marks · 2026, 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
occasional · NAT · 1 marks · 2018, 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
occasional · mixed · 2 marks · 2025, 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
common · NAT · 1 marks · 2026, 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
rare · NAT · 1 marks · 2014
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
common · MCQ · 1 marks · 2015
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
common · NAT · 1 marks · 2026
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
common · MSQ · 2 marks · 2024
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
rare · MCQ · 1 marks · 2016
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 (cA) = c (A) instead of cn (A) for an n n matrix.
Assuming (A+B) = (A) + (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 ( a1a2 = b1b2 c1c2) 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.