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Matrices and Determinants Class 12: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Q: What is a matrix in Class 12 Maths?

A matrix is a rectangular arrangement of numbers (or functions) in rows and columns. It is denoted by a capital letter and its order is written as m × n, where m is the number of rows and n is the number of columns. Matrices are used to represent linear transformations and solve systems of linear equations.

Q: What is the difference between a matrix and a determinant?

A matrix is a rectangular array of numbers, while a determinant is a scalar value that can be computed from a square matrix. Only square matrices have determinants. The determinant of a matrix A is denoted by |A| or det(A).

Q: Can I download the Matrices and Determinants formula sheet PDF for free?

Yes. You can download the complete Matrices and Determinants formula sheet PDF for free using the download button on this page. It covers types of matrices, matrix operations, determinant properties, adjoint, inverse, and solving linear equations in one comprehensive place for quick revision before JEE and board exams.

By Rohit Gupta Aug 24, 2026 13 min read

Matrices and Determinants Class 12: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Matrices and Determinants — Competishun

Matrices and Determinants Class 12: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Matrix Operations · Determinants · Adjoint · Inverse · Linear Equations

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Matrices and Determinants is one of the most important and scoring chapters in Class 12 Maths. It carries significant weightage in JEE and is a guaranteed source of marks in board exams. The chapter deals with the algebra of matrices, determinants, and their applications in solving linear equations.

This chapter covers types of matrices, matrix operations (addition, multiplication, transpose), determinants (properties, evaluation), adjoint, inverse of a matrix, and solving systems of linear equations using matrices (Cramer's rule and matrix method). Understanding these concepts is essential for success in linear algebra and beyond.

This page gives you the complete guide to Matrices and Determinants with all concepts explained in depth. You will find clear definitions, formulas, visual diagrams, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.

MatricesTypes · Operations · Transpose

DeterminantsProperties · Evaluation

InverseA⁻¹ = adj(A) / |A|

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What are Matrices and Determinants?

Definition: A matrix is a rectangular arrangement of numbers (or functions) in rows and columns. A determinant is a scalar value that can be computed from a square matrix. Only square matrices have determinants.

Matrices and determinants are used to represent linear transformations, solve systems of linear equations, and perform operations in linear algebra.

Matrices

A rectangular array of numbers arranged in rows and columns. Order is m × n (m rows, n columns).

Determinants

A scalar value associated with a square matrix. Denoted by |A| or det(A). Used to find inverses and solve linear equations.

Key Insight: Matrices provide a compact way to represent and operate on multiple linear equations. Determinants tell us whether a system of equations has a unique solution.

Glossary of Matrices and Determinants Terms — Complete A to Z

Before diving deep into each topic, let's understand the key terminology used in this chapter:

Term	Definition
Matrix	A rectangular arrangement of numbers in rows and columns.
Order of a Matrix	m × n, where m is the number of rows and n is the number of columns.
Square Matrix	A matrix with the same number of rows and columns (m = n).
Row Matrix	A matrix with only one row (1 × n).
Column Matrix	A matrix with only one column (m × 1).
Diagonal Matrix	A square matrix with non-zero elements only on the main diagonal.
Scalar Matrix	A diagonal matrix where all diagonal elements are equal.
Identity Matrix (I)	A diagonal matrix with all diagonal elements equal to 1.
Zero Matrix (O)	A matrix where all elements are zero.
Transpose of a Matrix (Aᵀ)	The matrix obtained by interchanging rows and columns of A.
Symmetric Matrix	A square matrix such that A = Aᵀ.
Skew-Symmetric Matrix	A square matrix such that A = −Aᵀ.
Determinant	A scalar value computed from a square matrix. Denoted by \|A\| or det(A).
Minor	The determinant of the submatrix formed by deleting the ith row and jth column.
Cofactor	Mᵢⱼ = (−1)ⁱ⁺ʲ × Minorᵢⱼ.
Adjoint	The transpose of the cofactor matrix.
Inverse Matrix	A⁻¹ = adj(A) / \|A\|, exists only if \|A\| ≠ 0.
Singular Matrix	A square matrix with determinant equal to zero.
Non-Singular Matrix	A square matrix with determinant not equal to zero.
Cramer's Rule	A method for solving linear equations using determinants.
Mastering these terms is essential for understanding Matrices and Determinants.

Types of Matrices

Definition: Matrices can be classified based on their order, elements, and properties. The table below lists the important types of matrices.

Row Matrix

Only one row (1 × n)

[1 2 3]

Column Matrix

Only one column (m × 1)

[1; 2; 3]

Square Matrix

m = n

[1 2; 3 4]

Diagonal Matrix

Non-zero only on main diagonal

diag(1, 2, 3)

Scalar Matrix

Diagonal with equal elements

diag(5, 5, 5)

Identity Matrix (I)

Scalar matrix with diagonal = 1

I₂ = [1 0; 0 1]

Zero Matrix (O)

All elements are zero

[0 0; 0 0]

Transpose (Aᵀ)

Rows ↔ Columns

(Aᵀ)ᵢⱼ = Aⱼᵢ

Symmetric

A = Aᵀ

[1 2; 2 3]

Skew-Symmetric

A = −Aᵀ

[0 a; −a 0]

Important: In a skew-symmetric matrix, all diagonal elements must be zero.

Matrix Operations — Addition, Multiplication, and Transpose

Definition: Matrix operations allow us to combine and manipulate matrices. The key operations are addition, subtraction, multiplication (by a scalar and by another matrix), and transpose.

Matrix Addition and Subtraction

Addition: A + B = [aᵢⱼ + bᵢⱼ] (same order required).
Subtraction: A − B = [aᵢⱼ − bᵢⱼ] (same order required).
Scalar Multiplication: kA = [k · aᵢⱼ].

Matrix Multiplication

Condition: Number of columns in A must equal number of rows in B.
Formula: (AB)ᵢⱼ = Σ aᵢₖ · bₖⱼ (sum over k).
Not commutative: AB ≠ BA in general.

Transpose of a Matrix

Definition: (Aᵀ)ᵢⱼ = Aⱼᵢ (interchange rows and columns).
Properties: (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (AB)ᵀ = BᵀAᵀ.

Matrix Multiplication

For matrix multiplication, the number of columns of A must equal the number of rows of B.

Key Insight: Matrix multiplication is not commutative. The order of multiplication matters. AB ≠ BA in general.

Determinants — Evaluation and Properties

Definition: The determinant of a square matrix A is a scalar value denoted by |A| or det(A). It is computed by expanding along a row or column using cofactors.

Determinant of a 2 × 2 Matrix

|A| = |a b; c d| = ad − bc

Determinant of a 3 × 3 Matrix

|A| = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)

Important Properties of Determinants

Product Property

|AB| = |A| · |B|

Determinant of product = product of determinants

Transpose Property

|Aᵀ| = |A|

Determinant unchanged by transpose

Scalar Multiple

|kA| = kⁿ|A|

n = order of matrix

Identical Rows/Columns

If two rows/columns are identical, |A| = 0

Singular matrix

Row/Column Swap

Swapping two rows/columns changes sign

Sign reversal

Row/Column Addition

Adding a multiple of one row/column to another leaves |A| unchanged

Used for simplification

Inverse Property

|A⁻¹| = 1 / |A|

For non-singular A

Product of Adjoint

|adj(A)| = |A|ⁿ⁻¹

n = order of matrix

Important: The determinant of a matrix is zero if and only if the matrix is singular (not invertible). This is a key test for the existence of an inverse.

Adjoint and Inverse of a Matrix

Definition: The adjoint of a matrix A (denoted adj(A)) is the transpose of the cofactor matrix. The inverse of a non-singular matrix A is given by A⁻¹ = adj(A) / |A|.

Steps to Find the Inverse

Step 1: Find the determinant |A|. If |A| = 0, the inverse does not exist.
Step 2: Find the cofactor matrix C (where Cᵢⱼ = (−1)ⁱ⁺ʲ · Mᵢⱼ).
Step 3: Transpose the cofactor matrix to get adj(A).
Step 4: A⁻¹ = adj(A) / |A|.

Important Relations

Inverse Formula

A⁻¹ = adj(A) / |A|

Valid only if |A| ≠ 0

Product of A and adj(A)

A · adj(A) = |A| · I

Where I is the identity matrix

Determinant of Adjoint

|adj(A)| = |A|ⁿ⁻¹

n = order of matrix

Adjoint of Adjoint

adj(adj(A)) = |A|ⁿ⁻² · A

For n ≥ 2

Key Insight: The inverse of a matrix is the matrix analogue of the reciprocal of a number. It is used to solve systems of linear equations and to find the solution of matrix equations.

Solving Linear Equations Using Matrices

Definition: A system of linear equations can be written in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Cramer's Rule

For a system of n linear equations in n variables, the solution is given by:

x₁ = |A₁| / |A|, x₂ = |A₂| / |A|, ..., xₙ = |Aₙ| / |A|
Where Aᵢ is the matrix obtained by replacing the ith column of A with B.

Matrix Method (Inverse Method)

If A is non-singular, the system AX = B has the unique solution:

X = A⁻¹B

Important: If |A| = 0 and (adj A)B = 0, the system has infinitely many solutions (consistent). If |A| = 0 and (adj A)B ≠ 0, the system has no solution (inconsistent).

Practice Questions — From JEE and Boards

Question	Answer
Q1: Find the determinant of A = [2 3; 4 5].	\|A\| = 2×5 − 3×4 = 10 − 12 = −2.
Q2: Find the transpose of A = [1 2; 3 4].	Aᵀ = [1 3; 2 4].
Q3: Is A = [1 2; 2 3] symmetric?	Yes, because A = Aᵀ.
Q4: Find the inverse of A = [2 1; 1 1].	\|A\| = 2×1 − 1×1 = 1. adj(A) = [1 -1; -1 2]. A⁻¹ = [1 -1; -1 2].
Q5: Find the determinant of A = [1 2 3; 4 5 6; 7 8 9].	\|A\| = 0 (rows are linearly dependent).
Q6: Solve the system: x + y = 5, 2x − y = 1 using Cramer's rule.	\|A\| = −3. \|A₁\| = −6, \|A₂\| = −9. x = 2, y = 3.
Q7: What is the condition for a matrix to be invertible?	\|A\| ≠ 0 (non-singular).
Q8: If A and B are square matrices of the same order, what is \|AB\|?	\|AB\| = \|A\| · \|B\|.
Practise these types of questions to become comfortable with applying Matrices and Determinants concepts in exam scenarios.

All Matrices and Determinants Formulas at a Glance

Formula	What It Means
A + B = [aᵢⱼ + bᵢⱼ]	Matrix addition
kA = [k · aᵢⱼ]	Scalar multiplication
(AB)ᵢⱼ = Σ aᵢₖ · bₖⱼ	Matrix multiplication
(Aᵀ)ᵢⱼ = Aⱼᵢ	Transpose
\|A\| = ad − bc (2×2)	Determinant of 2×2 matrix
\|A\| = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(...) + a₁₃(...) (3×3)	Determinant of 3×3 matrix
\|AB\| = \|A\| · \|B\|	Product property
\|Aᵀ\| = \|A\|	Transpose property
\|kA\| = kⁿ\|A\|	Scalar multiple property
A⁻¹ = adj(A) / \|A\|	Inverse formula
X = A⁻¹B	Solving linear equations
Cᵢⱼ = (−1)ⁱ⁺ʲ · Mᵢⱼ	Cofactor
Memorise these formulas for Matrices and Determinants. They are the key to scoring full marks in this chapter.

Common Mistakes in Matrices and Determinants

Confusing matrix multiplication with scalar multiplication: Matrix multiplication is not element-wise; it follows the row-column rule.
Forgetting that AB ≠ BA: Matrix multiplication is not commutative.
Confusing determinant and matrix: A determinant is a scalar; a matrix is an array.
Forgetting the sign in cofactor expansion: Cofactor = (−1)ⁱ⁺ʲ × Minor.
Misapplying the transpose property: (AB)ᵀ = BᵀAᵀ (order reverses).
Not checking if |A| = 0 before finding the inverse: Only non-singular matrices have inverses.

Golden Rule: In Matrices and Determinants, always check the order of matrices before performing operations. Use the correct formulas for determinants and inverses.

Why Matrices and Determinants Matters for JEE and Boards

Foundation for linear algebra: Matrices and determinants are the building blocks of linear algebra, which is used in many areas of mathematics and science.
High weightage: This chapter appears in 1-2 questions in JEE Main and is a guaranteed source of marks in board exams.
Conceptual clarity: This chapter rewards students who understand the definitions and properties rather than just memorizing formulas.
Practical relevance: Matrices and determinants are used in physics, engineering, computer science, and economics.

Why this guide helps: A comprehensive Matrices and Determinants guide with all concepts, definitions, formulas, and practice questions saves you time during revision and helps you quickly recall everything during the exam. You won't need to look anywhere else.

Get the Complete Matrices and Determinants PDF for Free

Download the full Matrices and Determinants guide with all concepts, definitions, formulas, and practice questions. Perfect for last-minute revision before JEE and board exams.

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Frequently Asked Questions — Matrices and Determinants

What is a matrix in Class 12 Maths?

A matrix is a rectangular arrangement of numbers (or functions) in rows and columns. It is denoted by a capital letter and its order is written as m × n, where m is the number of rows and n is the number of columns. Matrices are used to represent linear transformations and solve systems of linear equations.

What is the difference between a matrix and a determinant?

A matrix is a rectangular array of numbers, while a determinant is a scalar value that can be computed from a square matrix. Only square matrices have determinants. The determinant of a matrix A is denoted by |A| or det(A).

What are the important properties of determinants?

Key properties include: (1) det(AB) = det(A) · det(B), (2) det(Aᵀ) = det(A), (3) If two rows/columns are identical, det = 0, (4) If a row/column is multiplied by a scalar k, the determinant becomes k times the original, (5) det(kA) = kⁿ det(A) for an n × n matrix.

How do you find the inverse of a matrix?

The inverse of a square matrix A exists only if |A| ≠ 0. It is given by A⁻¹ = adj(A) / |A|, where adj(A) is the adjoint (transpose of the cofactor matrix). The inverse matrix satisfies A · A⁻¹ = A⁻¹ · A = I, where I is the identity matrix.

Can I download the Matrices and Determinants formula sheet PDF for free?

Yes. You can download the complete Matrices and Determinants formula sheet PDF for free using the download button on this page. It covers types of matrices, matrix operations, determinant properties, adjoint, inverse, and solving linear equations in one comprehensive place for quick revision before JEE and board exams.

Matrices and Determinants Class 12: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Download the Matrices and Determinants Complete Guide PDF

What are Matrices and Determinants?

Matrices

Determinants

Glossary of Matrices and Determinants Terms — Complete A to Z

Types of Matrices

Matrix Operations — Addition, Multiplication, and Transpose

Matrix Addition and Subtraction

Matrix Multiplication

Transpose of a Matrix

Determinants — Evaluation and Properties

Determinant of a 2 × 2 Matrix

Determinant of a 3 × 3 Matrix

Important Properties of Determinants

Product Property

Transpose Property

Scalar Multiple

Identical Rows/Columns

Row/Column Swap

Row/Column Addition

Inverse Property

Product of Adjoint

Adjoint and Inverse of a Matrix

Steps to Find the Inverse

Important Relations

Inverse Formula

Product of A and adj(A)

Determinant of Adjoint

Adjoint of Adjoint

Solving Linear Equations Using Matrices

Cramer's Rule

Matrix Method (Inverse Method)

Practice Questions — From JEE and Boards

All Matrices and Determinants Formulas at a Glance

Common Mistakes in Matrices and Determinants

Why Matrices and Determinants Matters for JEE and Boards

Get the Complete Matrices and Determinants PDF for Free