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Matrices and Determinants
Matrices and Determinants
Matrices and Determinants – JEE Mains Mathematics
by Zenith LMS
Updated April 2025
1. Matrices
A matrix is a rectangular array of numbers arranged in rows and columns.
Order of a matrix: m × n (m rows and n columns)
2. Types of Matrices
Square Matrix:
Number of rows = number of columns
Diagonal Matrix:
All non-diagonal elements are zero
Scalar Matrix:
A diagonal matrix with equal diagonal elements
Identity Matrix:
A diagonal matrix with all diagonal elements = 1
Zero Matrix:
All elements are zero
Row Matrix:
Only one row
Column Matrix:
Only one column
Symmetric and Skew-Symmetric Matrices
3. Algebra of Matrices
Addition and Subtraction (only possible if matrices are of same order)
Multiplication (A × B is defined only if the number of columns in A = number of rows in B)
Scalar multiplication and Transpose of a Matrix
4. Determinants
Defined only for square matrices
Determinants of 2×2 and 3×3 matrices
Used in finding area of triangle: If points are (x₁, y₁), (x₂, y₂), (x₃, y₃):
Area = (1/2) × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
5. Properties of Determinants
Interchanging rows/columns changes the sign
Two identical rows/columns ⇒ determinant = 0
If a row/column is multiplied by a scalar, determinant is also multiplied by the scalar
6. Adjoint and Inverse of a Matrix
Adjoint:
Transpose of the cofactor matrix
Inverse of matrix A:
A⁻¹ = adj(A)/|A| (only if |A| ≠ 0)
7. Solving System of Linear Equations
Using matrix method:
AX = B ⇒ X = A⁻¹B, where A is coefficient matrix, X is variable matrix, B is constant matrix
Applicable only if A⁻¹ exists (i.e., |A| ≠ 0)
Used for 2 or 3 variable systems
8. Consistency of System
Consistent:
Has at least one solution
Inconsistent:
No solution
Tested using rank or determinant
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JEE Mains Physics
Units and Measurements
Kinematics
Laws of Motion
Work, Energy and Power
Rotational Motion
Gravitation
Properties of Solids and Liquids
Thermodynamics
Kinetic Theory of Gases
Oscillations and Waves
Electrostatics
Current Electricity
Magnetic Effects of Current and Magnetism
Electromagnetic Induction and Alternating Currents
Electromagnetic Waves
Optics
Dual Nature of Matter and Radiation
Atoms and Nuclei
Electronic Devices
Experimental Skills
JEE Mains Chemistry
Some Basic Concepts in Chemistry
Atomic Structure
Chemical Bonding and Molecular Structure
Chemical Thermodynamics
Solutions
Equilibrium
Redox Reactions and Electrochemistry
Chemical Kinetics
Classification of Elements and Periodicity
p-Block Elements
d-Block and f-Block Elements
Coordination Compounds
Purification and Characterisation of Organic Compounds
Some Basic Principles of Organic Chemistry
Hydrocarbons
Organic Compounds Containing Halogens
Organic Compounds Containing Oxygen
Organic Compounds Containing Nitrogen
Biomolecules
Practical Chemistry Principles
JEE Mains Mathematics
Integral Calculus
Application of Derivatives
Continuity and Differentiability
Sets, Relations, and Functions
Complex Numbers Quadratics
Matrices and Determinants
Permutations and Combinations
Binomial Theorem and Its Simple Applications
Sequence and Series
Limit, Continuity, and Differentiability
Differential Equations
Coordinate Geometry
Three Dimensional Geometry
Vector Algebra
Statistics and Probability
Trigonometry