GATE Engineering Mathematics Syllabus

Engineering Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Engineering Mathematics in GATE, not a summary of it.

1. 1

   Linear Algebra

   3 topics

   • Matrices and Determinants
     • Matrix algebra and rank
     • System of linear equations and consistency
   • Eigenvalues and Eigenvectors
     • Characteristic equation and Cayley-Hamilton
     • Diagonalization
   • Vector Spaces
     • Linear independence and basis
     • Rank-nullity theorem
2. 2

   Calculus

   5 topics

   • Limits, Continuity and Differentiability
   • Mean Value Theorems and Taylor Series
     • Rolle's and Lagrange's theorems
     • Maxima and minima
   • Multivariable Calculus
     • Partial derivatives and gradient
     • Directional derivatives, divergence and curl
   • Integration
     • Definite and improper integrals
     • Double and triple integrals
     • Line, surface and volume integrals
   • Vector Calculus Theorems
     • Green's theorem
     • Stokes' and Gauss divergence theorems
3. 3

   Differential Equations

   4 topics

   • First Order ODEs
     • Separable, linear and exact equations
     • Integrating factors
   • Higher Order Linear ODEs
     • Homogeneous and particular solutions
     • Method of variation of parameters
   • Partial Differential Equations
     • Classification of second order PDEs
     • Heat, wave and Laplace equations
   • Initial and Boundary Value Problems
4. 4

   Probability and Statistics

   4 topics

   • Probability Fundamentals
     • Conditional probability and Bayes' theorem
     • Random variables and distributions
   • Common Distributions
     • Binomial, Poisson and normal
     • Exponential and uniform
   • Descriptive Statistics
     • Mean, median, mode and standard deviation
     • Correlation and regression
   • Sampling and Estimation
5. 5

   Numerical Methods and Complex Variables

   4 topics

   • Numerical Solution of Equations
     • Bisection, Newton-Raphson and secant methods
   • Numerical Integration
     • Trapezoidal and Simpson's rules
   • Numerical Solution of ODEs
     • Euler and Runge-Kutta methods
   • Complex Analysis
     • Analytic functions and Cauchy-Riemann equations
     • Cauchy's integral theorem and residues

Engineering Mathematics flashcards for GATE

21 of 50 cards from the Engineering Mathematics deck — real questions with worked answers.

1. What is the determinant of a $2\times 2$ matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

   $\det(A)=ad-bc$.

2. State the key property relating the determinant of a product of two square matrices $A$ and $B$.

   $\det(AB)=\det(A)\,\det(B)$.

3. How does $\det(A^{-1})$ relate to $\det(A)$, and when does $A^{-1}$ exist?

   $\det(A^{-1})=\dfrac{1}{\det(A)}$, and $A^{-1}$ exists if and only if $\det(A)\neq 0$ (i.e. $A$ is non-singular).

4. If $A$ is an $n\times n$ matrix and $k$ a scalar, what is $\det(kA)$?

   $\det(kA)=k^{n}\det(A)$.

5. Give the formula for the inverse of an invertible matrix $A$ in terms of its adjugate.

   $A^{-1}=\dfrac{1}{\det(A)}\,\operatorname{adj}(A)$, where $\operatorname{adj}(A)$ is the transpose of the cofactor matrix.

6. Define the rank of a matrix.

   The rank is the maximum number of linearly independent rows (equivalently columns), i.e. the order of the largest non-zero minor, or the number of non-zero rows in its row-echelon form.

7. For an $m\times n$ matrix $A$, what is the maximum possible value of $\operatorname{rank}(A)$?

   $\operatorname{rank}(A)\leq \min(m,n)$.

8. What does it mean for an $n\times n$ matrix to have full rank, and what does it imply about invertibility?

   Full rank means $\operatorname{rank}(A)=n$. This implies $\det(A)\neq 0$, so $A$ is invertible (non-singular).

9. State the rank condition (Rouché–Capelli theorem) for consistency of the linear system $AX=B$.

   The system is consistent if and only if $\operatorname{rank}(A)=\operatorname{rank}([A\,|\,B])$ (the coefficient and augmented matrices have equal rank).

10. For a consistent system $AX=B$ in $n$ unknowns, when is the solution unique versus infinitely many?

    Unique solution if $\operatorname{rank}(A)=\operatorname{rank}([A|B])=n$; infinitely many solutions if $\operatorname{rank}(A)=\operatorname{rank}([A|B])=r<n$ (with $n-r$ free parameters).

11. For a homogeneous system $AX=0$ in $n$ unknowns, when does it have a non-trivial solution?

    When $\operatorname{rank}(A)=r<n$, giving $n-r$ free variables. For a square $A$ this means $\det(A)=0$.

12. Define an eigenvalue and eigenvector of a square matrix $A$.

    A scalar $\lambda$ is an eigenvalue with eigenvector $\vec{x}\neq \vec{0}$ if $A\vec{x}=\lambda\vec{x}$.

13. Write the characteristic equation used to find the eigenvalues of $A$.

    $\det(A-\lambda I)=0$.

14. What is the relationship between the trace of $A$ and its eigenvalues?

    $\operatorname{tr}(A)=\sum_{i}\lambda_{i}$, i.e. the sum of eigenvalues equals the sum of the diagonal entries.

15. What is the relationship between $\det(A)$ and its eigenvalues?

    $\det(A)=\prod_{i}\lambda_{i}$, i.e. the product of all eigenvalues.

16. If $\lambda$ is an eigenvalue of $A$, what are the eigenvalues of $A^{2}$, $A^{-1}$, and $A^{k}$?

    $\lambda^{2}$ for $A^{2}$, $\dfrac{1}{\lambda}$ for $A^{-1}$ (if $\lambda\neq 0$), and $\lambda^{k}$ for $A^{k}$, with the same eigenvectors.

17. State the eigenvalue property of the eigenvalues of a triangular matrix.

    The eigenvalues of a triangular (or diagonal) matrix are exactly its diagonal entries.

18. State the Cayley–Hamilton theorem.

    Every square matrix satisfies its own characteristic equation: if $p(\lambda)=\det(A-\lambda I)$, then $p(A)=0$.

19. How can the Cayley–Hamilton theorem be used to find $A^{-1}$ for a non-singular matrix?

    From its characteristic equation $A^{n}+c_{n-1}A^{n-1}+\cdots+c_{1}A+c_{0}I=0$, solve for $I$ to express $A^{-1}=-\dfrac{1}{c_{0}}\big(A^{n-1}+c_{n-1}A^{n-2}+\cdots+c_{1}I\big)$.

20. When is a square matrix $A$ diagonalizable in terms of its eigenvectors?

    $A$ is diagonalizable if and only if it has $n$ linearly independent eigenvectors (e.g. when all $n$ eigenvalues are distinct, or each eigenvalue's geometric multiplicity equals its algebraic multiplicity).

21. If $A$ is diagonalizable, write the diagonalization formula and what $P$ and $D$ contain.

    $A=PDP^{-1}$, where the columns of $P$ are the eigenvectors and $D$ is the diagonal matrix of corresponding eigenvalues.

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