Linear Algebra Questions in GATE CSE: Eigenvalues to LU

Linear algebra gives GATE CSE its single most dependable question. Eigenvalue identities score 94.0 out of 100 in the book's evidence model, the joint highest in the mathematics section, and the reconstructed record puts an eigenvalue item in essentially every paper since 2003. Four question types carry the topic, and each one is a short computation rather than a proof.

Key takeaways

• The syllabus names matrices, determinants, systems of linear equations, eigenvalues and eigenvectors, and LU decomposition. Nothing else is in scope.
• Eigenvalue questions are solved with trace and determinant identities, not with characteristic polynomials.
• Rank-based reasoning has replaced mechanical Gaussian elimination in recent papers.
• Determinant questions test properties. Only products and polynomials of one matrix behave predictably; sums do not.
• LU decomposition scores 54.5 and appears rarely, but it is explicitly named and takes an evening to prepare.

The four question types

Years are an analyst reconstruction from public question archives and are approximate. Scores are model based analytical confidence values, not probabilities of appearance in 2027.

Type 1: eigenvalue identities

Almost no GATE eigenvalue question wants you to expand a characteristic polynomial. What they want is a derived quantity, and the identities give it in seconds.

Memorise six facts. The sum of the eigenvalues is the trace and their product is the determinant. If $A$ has eigenvalue $\lambda$, then:

Triangular matrices carry their eigenvalues on the diagonal.

Worked example: a $3 \times 3$ matrix $A$ has eigenvalues 1, 2 and 4. What is $\det(A^2 + 3I)$? The eigenvalues of that combination are $\lambda^2 + 3$:

For $2 \times 2$ matrices the reverse route is just as common. Given a trace of 7 and a determinant of 12, the eigenvalues solve $\lambda^2 - 7\lambda + 12 = 0$, a quadratic with sum 7 and product 12, so $\lambda = 3$ and $4$. No elimination required.

Special matrices give their eigenvalues away by type: orthogonal matrices have eigenvalues with $\lvert \lambda \rvert = 1$, idempotent matrices ($A^2 = A$) have only 0 and 1, nilpotent matrices have only 0, and involutory matrices ($A^2 = I$) have only $\pm 1$. Real symmetric matrices have real eigenvalues and orthogonal eigenvectors, which is the most quoted statement in multiple-select items on this topic.

Type 2: rank, nullity and systems of equations

The pattern that has survived two decades is the parameter question: for which values of a parameter does a $3 \times 3$ system have a unique solution, infinitely many, or none? Compare the rank of the coefficient matrix with the rank of the augmented matrix.

Take the system

Subtracting the first row from the other two gives $y + 2z = 4$ and $y + (a - 1)z = b - 6$. Subtracting again gives $(a - 3)z = b - 10$. So the system has a unique solution whenever $a \ne 3$, infinitely many solutions when $a = 3$ and $b = 10$, and no solution when $a = 3$ and $b \ne 10$.

The trap is stopping at the determinant. A zero determinant only tells you that the unique case has failed; the augmented rank decides between infinitely many and none.

Recent papers have moved toward rank facts rather than elimination. The two worth carrying are $\operatorname{rank}(AA^T) = \operatorname{rank}(A)$ and $\text{nullity}(A) = n - \operatorname{rank}(A)$, where $n$ is the number of columns. Homogeneous systems always have the zero solution and have exactly that many independent solutions.

The book ranks all 25 Engineering Mathematics concepts on the same evidence model and pairs each of these four question types with practice questions covering its recurring variants.

Type 3: determinants and matrix properties

Determinant questions test properties rather than arithmetic. Long expansions of $3 \times 3$ and $4 \times 4$ matrices have largely disappeared from the recent papers, replaced by property-based reasoning that a multiple-select question can grade.

The list that covers nearly every past item, for an $n \times n$ matrix $A$:

For example, if a $4 \times 4$ matrix has determinant 3, then $\det(2A) = 2^4 \times 3 = 48$. Answering 6 is the standard slip.

The counterpart trap is worth stating separately because it appears in both determinant and eigenvalue items: $\det(A + B) \ne \det A + \det B$ in general, and the eigenvalues of $A + B$ are not the sums of their eigenvalues. Only operations on a single matrix behave.

Cayley-Hamilton is the bridge between the two types. Every matrix satisfies its own characteristic equation, so for a $2 \times 2$ matrix with trace $t$ and determinant $d$, $A^2 = tA - dI$. That rearranges into an inverse expressed as a polynomial in the matrix, which is exactly what the paper asks for when it wants a matrix power or inverse without elimination.

Type 4: LU decomposition

LU decomposition is the clearest case in this subject of a topic that is explicitly named but rarely asked. It sits in the Moderate band at 54.5 because its history is thin and its variants are few, not because it is hard.

What has been asked is narrow: an entry of $L$ or $U$ for a small matrix, the existence condition, or the determinant read off the diagonal of $U$. Take the $2 \times 2$ matrix with first row 2, 3 and second row 4, 11. Under the Doolittle convention, where $L$ has ones on its diagonal, the multiplier is $4/2 = 2$:

The determinant is the product of the diagonal of $U$:

The one thing to fix in your notes is the convention. Doolittle puts the unit diagonal on $L$, Crout puts it on $U$, and a question that does not say which one it wants will tell you through the entries it gives you.

Where linear algebra fits in your plan

Linear algebra is part of the roughly 13 marks the official pattern page allots to Engineering Mathematics, shared with discrete mathematics, calculus and probability. Analyst compilations separating the two halves between 2016 and 2026 put the linear algebra, calculus and probability group between 4 and 14 marks, with a median near 6. Those figures are approximate and differ by about 2 to 3 marks between sources.

Practically, this means one eigenvalue item and one rank or systems item in most shifts. Prepare eigenvalues to the point of reflex, rank conditions to the point of a written checklist, determinant properties as a list you can recite, and LU to the point of recognising the convention. The priority order across the whole section is in Engineering Mathematics important topics, the formulas are on the GATE CSE formula sheet, the slips are catalogued in the common mistakes guide, and the method behind the rankings is explained in what counts as an important question.