Last Minute Notes Engineering Mathematics (original) (raw)

Last Updated : 20 Feb, 2026

GATE CSE is a national-level engineering entrance exam in India specifically for Computer Science and Engineering. It's conducted by top Indian institutions like IISc Bangalore and various IITs. In GATE CSE, engineering mathematics is a significant portion of the exam, typically constituting 15% of the total marks. Key topics in engineering mathematics tested in the exam include:

Check Complete **Last Minute Notes for GATE CSE.

Linear Algebra

Matrices

Example-of-Matrix

This Matrix [M] has 3 rows and 2 columns i.e., order of 3 × 2. Each element of matrix [M] can be referred to by its row and column number. For example, a32 = 0.

**Types of Matrices

**Operations on Matrices

Matrix Operations are basic calculations performed on matrices to solve problems or manipulate their structure.

**Transpose of a Matrix: The transpose [M]T of an m x n matrix [M] is the n × m matrix obtained by interchanging the rows and columns of [M]. if A= [aij] m × n , then AT = [bij] n × m where bij = aji

**Properties of transpose of a Matrix:

**Adjoint of a Matrix: The adjoint of a square matrix is the transpose of its cofactor matrix.

Adjoint-of-Matrix

Read More about **Minor and Cofactors.

**Properties of Adjoint

Some important properties of adjoint include:

**Inverse of a Matrix

For any square matrix A,

A^{-1} = \frac{Adj A}{|A|}

Here |A| should not be equal to zero, means matrix A should be non-singular.

**Properties of Inverse

**Note: Only a non-singular square matrix can have an inverse.

**Conjugate of a Matrix: If A is a matrix with elements aij​, then the conjugate matrix \overline{A} is obtained by taking the complex conjugate of each element aij​.

Mathematically conjugate of m × n matrix is given by: \overline{A} = \begin{bmatrix} \overline{a_{11}} & \overline{a_{12}} & \cdots & \overline{a_{1n}} \\ \overline{a_{21}} & \overline{a_{22}} & \cdots & \overline{a_{2n}} \\ \vdots & \vdots & \ddots & \vdots \\ \overline{a_{m1}} & \overline{a_{m2}} & \cdots & \overline{a_{mn}} \end{bmatrix}

Here, \overline{a_{ij}}​​ represents the complex conjugate of aij.

**Trace of a Matrix: Let A = [aij]n × n is a square matrix of order n, then the sum of diagonal elements is called the trace of a matrix which is denoted by tr(A).

**tr(A) = a 11 + a 22 + a 33 + ……….+ a nn

Remember trace of a matrix is also equal to the sum of eigen value of the matrix. For example:

Trace-of-Matrix

**Properties of Trace of Matrix:

Let A and B be any two square matrices of order n, then

  1. tr(kA) = k tr(A) where k is a scalar.
  2. tr(A+B) = tr(A)+tr(B)
  3. tr(A-B) = tr(A)-tr(B)
  4. tr(AB) = tr(BA)

**Determinant of Matrices

Determinant represents the scaling factor of the linear transformation associated with the matrix. For example, in a 2 × 2 matrix, the determinant represents the area scaling factor.

**Properties of Determinant

**Rank of a Matrix: Rank of matrix is the number of non-zero rows in the row reduced form or the maximum number of independent rows or the maximum number of independent columns. Rank is denoted as rank(A) or ρ(A). if A is a non-singular matrix of order n, then rank of A = n i.e. ρ(A) = n.

Let A be any m × n matrix and it has square sub-matrices of different orders. A matrix is said to be of rank r, if it satisfies the following properties:

**Properties of Rank of a Matrix

**Solution of a System of Linear Equations

Linear equations can have three kind of possible solutions:

**System of homogeneous linear equations AX = 0.

  1. X = 0. is always a solution; means all the unknowns has same value as zero. (This is also called trivial solution)
  2. If ρ(A) = number of unknowns, unique solution.
  3. If ρ(A) < number of unknowns, infinite number of solutions.

**System of non-homogeneous linear equations AX = B.

  1. If ρ[A:B] ≠ ρ(A), No solution.
  2. If P[A:B] = ρ(A) = the number of unknown variables, unique solution.
  3. If ρ[A:B] = ρ(A) ≠ number of unknown, infinite number of solutions.

Here ρ[A:B] is rank of gauss elimination representation of AX = B.

There are two states of the Linear equation system:

**Linear dependence and Linear independence of Vector:

**How to Determine Linear Dependency and Independency

Let X1, X2 ….Xr be the given vectors. Construct a matrix with the given vectors as its rows.

  1. If the rank of the matrix of the given vectors is less than the number of vectors, then the vectors are linearly dependent.
  2. If the rank of the matrix of the given vectors is equal to the number of vectors, then the vectors are linearly independent.

Eigen Value and Eigen Vector

Eigen vector of a matrix A is a vector represented by a matrix X such that when X is multiplied with matrix A, then the direction of the resultant matrix remains the same as vector X.

Mathematically, above statement can be represented as:

**AX = λX

Where A is any arbitrary matrix, λ are eigen values and X is an eigen vector corresponding to each eigen value.

Here, we can see that AX is parallel to X. So, X is an eigen vector.

**How to Find Eigenvalues and Eigen Vectors of Given Matrices

We know that,

AX = λX
AX – λX = 0
(A – λI) X = 0 . . . (1)

Above condition will be true only if (A – λI) is singular. That means,

|A – λI| = 0 . . . (2)

This is known as characteristic equation of the matrix and the roots of the characteristic equation are the eigen values of the matrix A.

**Properties of Eigen Values

LU Decomposition

**LU Decomposition (also known as LU Factorization) is a method used to solve a system of linear equations. It decomposes a given square matrix **A into the product of two matrices **L (Lower Triangular Matrix) and **U (Upper Triangular Matrix), such that:

A = L ⋅ U

Where:

Probability and Statistics

Probability

Probability refers to the extent of occurrence of events. When an event occurs like throwing a ball, picking a card from deck, etc ., then the must be some probability associated with that event.

**Basic Terminologies:

**Sample Space (S): The set of all possible outcomes of an experiment.

**Event (E): Any subset of the sample space.

**Probability of an Event (P(E)): A measure of the likelihood of an event occurring, where 0≤P(E)≤10 \leq P(E) \leq 10≤P(E)≤1.

**Important Rules:

**Addition Rule:

**Multiplication Rule:

**Events in Probaility

Events are subsets of a sample space and represent the outcomes or collections of outcomes of a random experiment.

**Types of Events

**Theorems: General - Let A, B, C are the events associated with a random experiment, then

**Extension of Multiplication Theorem: Let A1, A2, . . . , An are n events associated with a random experiment, then P(A1 ∩ A2 ∩ A3 . . . ∩ An) = P(A1)P(A2/A1)P(A3/A2∩A1) . . . P(An/A1∩A2∩A3∩ . . . ∩An-1)

**Total Law of Probability: Let S be the sample space associated with a random experiment and E1, E2, . . . , En be n mutually exclusive and exhaustive events associated with the random experiment . If A is any event which occurs with E1 or E2 or . . . or En, then

P(A) = P(E1)P(A/E1) + P(E2)P(A/E2) + ... + P(En)P(A/En)

**Conditional Probability: Conditional probability P(A | B) indicates the probability of event 'A' happening given that event B happened.

P(A|B) = \frac{P(A \cap B)}{P(B)}

**Product Rule: Derived from above definition of conditional probability by multiplying both sides with P(B)

P(A ∩ B) = P(B) × P(A|B)

**Bayes's Formula****:** If A1, A2, . . . , An are mutually exclusive and exhaustive events:

P(A_k|B) = \frac{P(B|A_k) P(A_k)}{\sum_{i=1}^n P(B|A_i) P(A_i)}

**Random Variables

A random variable is basically a function which maps from the set of sample space to set of real numbers.

**Discrete Random Variable: Takes finite or countably infinite values.

**Example: Number of heads in n coin tosses.

**Continuous Random Variable: Takes values in an interval.

**Example: Time taken to complete a task.

**Probability Distribution

A Probability Distribution describes how probabilities are assigned to outcomes or ranges of outcomes for a random variable.

**Probability Mass Function (PMF)

Used for discrete random variables.

**Probability Density Function (PDF)

**Cumulative Distribution Function (CDF)

For both discrete and continuous random variables.

F(x) = P(X ≤ x)

**Expected Value (Mean):

**Important Distributions

**Binomial Distribution: Binomial Distribution is used to calculate the probability of a specific number of successes in a fixed number of independent trials

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \dots, n

Mean: E[X] = np, Variance: Var(X) = np(1 − p).

**Poisson Distribution: Poisson Distribution is a discrete probability distribution that models the number of events occurring

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots

Mean and Variance: E[X] = Var(X) = λ.

**Uniform Distribution:

f(x) = 1/(b−a), a ≤ x ≤ b

Mean: E[X] = (a + b)/2, Variance: Var(X) = (b − a)2/12.

**Normal Distribution:

f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

68% of values lie within μ ± σ, 95% within μ ± 2σ.

**Exponential Distribution: For a positive real number \lambda the probability density function of a Exponentially distributed Random variable is given by:

f_X(x) =\begin{cases} \lambda e^{-\lambda x} & if x\in R_X \\ 0 & if x \notin R_X \end{cases}

Where Rx is exponential random variables.

Mean = 1/λ and Variance = 1/λ2

Statistics

Descriptive Statistics: It is a simple tools that help us understand and summarize data.

**Measures of Central Tendency:

**Measures of Spread:

**Inferential Statistics

Calculus

**Existence of Limit: The **limits of a function f(x) at x = a exists only when its left hand limit and right hand limit exist and are equal i.e. \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x)

Also Read **Formal Definition of Limit.

**Some Common Limits:

\bullet\: \lim_{x\to 0} \frac{\sin x}{x} = 1 \hspace{0.5cm} \\\bullet\: \lim_{x\to 0} \cos x = 1 \hspace{0.5cm} \\\bullet\: \lim_{x\to 0} \frac{\tan x}{x} = 1 \\ \bullet\: \lim_{x\to 0} \frac{1-\cos x}{x} = 0 \hspace{0.5cm}\\ \bullet\: \lim_{x\to 0} \frac{\sin x^\circ}{x} = \frac{\pi}{180} \hspace{0.5cm} \\\bullet\: \lim_{x\to a} \frac{x^n - a^n}{x-a} = na^{n-1} \\ \bullet\: \lim_{x\to \infty} \left(1+\frac{k}{x}\right)^{mx} = e^{mk} \hspace{0.5cm}\\ \bullet\: \lim_{x\to 0} (1+x)^{\frac{1}{x}} = e \hspace{0.5cm} \\\bullet\: \lim_{x\to 0} \frac{a^x-1}{x} = \ln a \\ \bullet\: \lim_{x\to 0} \frac{e^x-1}{x} = 1 \hspace{0.5cm} \\\bullet\: \lim_{x\to 0} \frac{\ln (1+x)}{x} = 1 \hspace{0.5cm} \\\bullet\: \lim_{x\to \infty} x^{\frac{1}{x}} = 1\\
Read More about **Properties of Limits.

**L'Hospital Rule:

If the given limit \lim_{x\to a} \frac{f(x)}{g(x)} is of the form \frac{0}{0} or \frac{\infty}{\infty} i.e. both f(x) and g(x) are either 0 or ∞, then the limit can be solved by **L'Hospital Rule.

If the limit is of the form described above, then the L'Hospital Rule says that:

\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f^\prime(x)}{g^\prime(x)}

Where f'(x) and g'(x) obtained by differentiating f(x) and g(x). If after differentiating, the form still exists, then the rule can be applied continuously until the form is changed.

**Continuity:

A function is said to be continuous over a range if it's graph is a single unbroken curve. Formally, a real valued function f(x) is said to be continuous at a point x = x0 in the domain if: \lim_{x\to x_\circ} f(x) exists and is equal to f(x0).

If a function f(x) is continuous at x = x0 then:

\lim_{x\to x_\circ ^+} f(x) = \lim_{x\to x_\circ ^-} f(x) = \lim_{x\to x_\circ} f(x)

Functions that are not continuous are said to be discontinuous.

Also Read about **Continuity at a Point.

**Differentiability:

The derivative of a real valued function f(x) wrt x is the function f'(x) and is defined as:

\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}

A function is said to be differentiable if the derivative of the function exists at all points of its domain. For checking the differentiability of a function at point x = c, \lim_{h\to 0} \frac{f(c+h)-f(c)}{h}
must exist.

**Note: If a function is differentiable at a point, then it is also continuous at that point, but if a function is continuous at a point does not imply that the function is also differentiable at that point. For example, f(x) = |x| is continuous at x = 0 but it is not differentiable at that point.

**Lagrange’s Mean Value Theorem:

**Suppose f:[a,b]\rightarrow R be a function satisfying three conditions:

Then according to Lagrange's Theorem, there exists at least one point 'c' in the open interval (a, b) such that:

f'(c)=\frac{f(b)-f(a)}{b-a}

**Rolle’s Mean Value Theorem:

Suppose f(x) be a function satisfying three conditions:

Then according to Rolle's Theorem, there exists at least one point 'c' in the open interval (a, b) such that:

f '(c) = 0

**Differentiation Formulas

Some of the most common formula used to find derivative are tabulated below:

d/dx(c) 0
d/dx{c.f(x)} c.f'(x)
d/dx(x) 1
d/dx(xn) nxn-1
d/dx{f(g(x))} f'(g(x)).g'(x)
d/dx(ax) ax.ln(a)
d/dx{ln(x)} {Note: ln(x) = loge(x)} 1/x, x>0
d/dx(logax) 1/xln(a)
d/dx(ex) ex
d/dx{sin(x)} cos(x)
d/dx{cos(x)} -sin(x)
d/dx{tan(x)} sec2x
d/dx{sec(x)} sec(x).tan(x)
d/dx{cosec(x)} -cosec(x).cot(x)
d/dx{cot(x)} -cosec2(x)
d/dx{sin-1(x)} 1/√(1 - x2)
d/dx{cos-1(x)} -1/√(1 - x2)
d/dx{tan-1(x)} 1/(1+x2)

Maxima and Minima

Read More about **Maxima and Minima.

**First Derivative Test****:**

If f′(x) changes:

**Second Derivative Test:

Compute the second derivative f′′(x):

**Concavity:

f(x) is:

Point of inflection: f(x) changes concavity (where f′′(x) = 0.

**Maxima and Minima in Multivariable Functions****:**

For f(x, y):

**Critical Points: Solve ∂f/∂x = 0 and ∂f/∂y = 0.

**Second **Partial Derivatives: Compute:

**Hessian Determinant: H = fxxfyy − (fxy)2.

**Integrals

Integrals can be classified as:

**Indefinite Integrals:

Let f(x) be a function. Then the family of all its antiderivatives is called the indefinite integral of a function f(x) and it is denoted by ∫f(x)dx.

**Fundamental Integration Formulas:

Some common integration formulas include:

  1. ∫xndx = (xn+1/(n+1))+C
  2. ∫(1/x)dx = (loge|x|)+C
  3. ∫exdx = (ex)+C
  4. ∫axdx = ((ax)/(logea))+C
  5. ∫sin(x)dx = -cos(x)+C
  6. ∫cos(x)dx = sin(x)+C
  7. ∫sec2(x)dx = tan(x)+C
  8. ∫cosec2(x)dx = -cot(x)+C
  9. ∫sec(x)tan(x)dx = sec(x)+C
  10. ∫cosec(x)cot(x)dx = -cosec(x)+C
  11. ∫cot(x)dx = log|sin(x)|+C
  12. ∫tan(x)dx = log|sec(x)|+C
  13. ∫sec(x)dx = log|sec(x)+tan(x)|+C
  14. ∫cosec(x)dx = log|cosec(x)-cot(x)|+C

**Definite Integrals:

Definite integrals are the extension after indefinite integrals, definite integrals have limits [a, b]. It gives the area of a curve bounded between given limits.

\int_{a}^{b}F(x)dx, it denotes the area of curve F(x) bounded between a and b, where a is the lower limit and b is the upper limit.

**Note: If f is a continuous function defined on the closed interval [a, b] and F be an anti derivative of f. Then \int_{a}^{b}f(x)dx= \left [ F(x) \right ]_{a}^{b} = F(a) - F(b).

Here, the function f needs to be well defined and continuous in [a, b].

  1. \int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt
  2. \int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx
  3. \int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx
  4. \int_{a}^{b}f(x)=\int_{a}^{b}f(a+b-x)dx
  5. \int_{0}^{b}f(x)=\int_{0}^{b}f(b-x)dx
  6. \int_{0}^{2a}f(x)dx=\int_{0}^{a}f(x)dx+\int_{0}^{a}f(2a-x)dx
  7. \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx, if f(x) is even function i.e f(x) = f(-x)
  8. \int_{-a}^{a}f(x)dx=0, if f(x) is odd function

**Newton-Leibnitz Rule

For a definite integral F(x) = \int_{a(x)}^{b(x)} f(t) \, dt:

\frac{d}{dx} \left[ \int_{a(x)}^{b(x)} f(t) \, dt \right] = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)

Application of Integrals

**Area Under a Curve

The area enclosed between a curve y = f(x), the x-axis, and the limits x = a and x = b is:

\text{Area} = \int_a^b f(x) \, dx

**Between Two Curves

The area between two curves y = f(x) and y = g(x) from x = a to x = b is:

Area = \int_a^b \big| f(x) - g(x) \big| \, dx.

**Length of a Curve

The length of a curve y = f(x) from x = a to x = b is:

Length = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx

For parametric equations x = x(t), y = y(t), the arc length is:

Length = \int_{t_1}^{t_2} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt

**Volume of Solids of Revolution

**Disk Method: When a curve y = f(x) is revolved about the x-axis: Volume = \pi \int_a^b \left[ f(x) \right]^2 \, dx

**Shell Method: When a curve y = f(x) is revolved about the y-axis: Volume = 2\pi \int_a^b x \cdot f(x) \, dx.

For parametric equations x = x(t), y = y(t):

**Surface Area of Solids of Revolution

  1. **Revolution about the x-axis:Surface Area = 2\pi \int_a^b f(x) \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
  2. **Revolution about the y-axis: Surface Area = 2\pi \int_a^b x \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx