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Class 12 Chapter 3

Class 12 Maths Chapter 3: Matrices

From computer graphics and artificial intelligence to cryptography, economics, and engineering, Matrices provide a powerful mathematical framework for organizing and solving complex problems involving multiple variables. This chapter introduces the concepts of types of matrices, matrix operations, transpose, symmetric and skew-symmetric matrices, elementary operations, and invertible matrices. As one of the most application-oriented topics in Mathematics, Matrices form the backbone of Linear Algebra and play a significant role in JEE Main, JEE Advanced, and CBSE Board examinations.

At ALLEN, our expert faculty have designed these Class 12 Maths Chapter 3: Matrices Revision Notes to help students revise the chapter efficiently with complete conceptual clarity. These notes include concise theory, important formulas, matrix properties, shortcut techniques, solved illustrations, common mistakes, and PYQ-based insights. Whether you're preparing for CBSE Board exams or competitive exams like JEE Main and JEE Advanced, these revision notes will strengthen your fundamentals and improve problem-solving accuracy.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	20–25 Minutes
Key Concepts Covered	Types of Matrices, Matrix Operations, Invertible Matrices

Important topic/subtopics covered:-

Topic	Overview
Matrix	Definition and notation
Types of Matrices	Row, Column, Square, Diagonal, Identity, Zero, Scalar, Symmetric, Skew-Symmetric
Matrix Operations	Addition, subtraction, scalar multiplication, multiplication
Transpose of a Matrix	Definition and properties
Symmetric & Skew-Symmetric Matrices	Characteristics and applications
Elementary Operations	Row and column transformations
Invertible Matrices	Definition and basic properties
Applications	Linear equations, transformations, and higher mathematics

2.0Chapter 3 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Matrices

A Matrix is an organized rectangular array comprising numbers, symbols, or expressions that are systematically arranged in rows and columns. It is typically denoted by uppercase letters and is used in Mathematics and Computer science to represent data or perform operations such as addition, subtraction, multiplication, and more.

1.0Types of Matrices

Column Matrix

A matrix is termed a Column Matrix if it contains only one column.

Example: $A = 03 - 2 \frac{1}{2}$

Row Matrix

A matrix is considered a Row Matrix if it consists of only one row.

Example - A = [1 5 7]

Square Matrix

A square matrix is one where the number of rows equals the number of columns.

Example : $A = [5173]$

Diagonal Matrix

A square matrix B = [b_ij] of order m × m is classified as a Diagonal Matrix if all its non-diagonal elements are zero. In other words, a matrix B = [b_ij] of size m × m is designated as a diagonal matrix if b_ij = 0 whenever i ≠ j.

$1000050000700003$

Scalar Matrix

A Scalar Matrix is a square matrix where all diagonal elements are equal (scalar), and all non-diagonal elements are zero.

Identity Matrix

A square matrix in which ones are on the main diagonal and zeros anywhere else is known as an identity matrix. It is denoted as I_n for an n × n matrix.

$100010001$

Zero Matrix

A zero matrix, denoted as 0, is a matrix in which all elements are zero.

$000000000$

Orthogonal Matrix

An orthogonal matrix is a square matrix where the transpose is equal to its inverse.

A^T = A^–1

$cos θ - sin θ 0 - sin θ cos θ 0 001$

Also Read Types of Matrices

1.0Operation on Matrices

Addition of two Matrices

To add two matrices, you add corresponding elements in the same position in each matrix. Here is the general formula for adding two matrices:

If two matrices A = [a_ij] and B=[b_ij] have same order m×n, then their sum A + B is given by:

A + B=[a_ij] + [b_ij] = [a_ij +b_ij]

Example:

$A = [475839] B = [716223]$

$A + B = [4 + 7 7 + 1 5 + 6 8 + 2 3 + 2 9 + 3] = [1181110512]$

Addition of matrices follows the following Properties
Commutative Law: A + B = B + A
Associative Law: (A + B) + C = A + (B + C)
Additive Identity: O (Zero Matrix) is the Additive Identity for matrix addition.
Additive Inverse: –A is the additive inverse of A or negative of A.

Multiplication of a matrix by a scalar

Multiplying a matrix by a scalar involves multiplying each element of the matrix by the scalar value. For example, if we have a matrix A and a scalar k, the product kA is obtained by multiplying each element of A by k.

Example:

$A = [475839]$

$kA = k \times [475839] = [4 k 7 k 5 k 8 k 3 k 9 k]$

Scalar Multiplication of a matrix follows the following Properties
k(A +B) = k A + kB,

Multiplication of Matrices

Matrix multiplication is a binary operation that combines two matrices to produce a new matrix. For matrix multiplication to be defined the number of columns in matrix A should match the number of rows in matrix B.

Let's consider two matrices, A and B:

$A = a_{11} a_{21} a_{31} a_{12} a_{22} a_{32}$

$B = [b_{11} b_{21} b_{12} b_{22} b_{13} b_{23}]$

To perform matrix multiplication between matrices A and B, we compute each element of the resulting matrix C by taking the scalar product of the corresponding row of matrix A and the respective column of matrix B:

To multiply A and B, we follow the rule that the (i,j)^th element of the resulting matrix C is obtained by taking the scalar product of the i^th row of A and the j^th column of B:

$C = A \cdot B = c_{11} c_{21} c_{31} c_{12} c_{22} c_{32} c_{13} c_{23} c_{33}$

where

c₁₁ = a₁₁⋅b₁₁ + a₁₂⋅b₂₁,

c₁₂ = a₁₁⋅b₁₂ + a₁₂⋅b₂₂,

c₁₃ = a₁₁⋅b₁₃ + a₁₂⋅b₂₃,

c₂₁ = a₂₁⋅b₁₁ + a₂₂⋅b₂₁,

c₂₂ = a₂₁⋅b₁₂ + a₂₂⋅b₂₂,

c₂₃ = a₂₁⋅b₁₃ + a₂₂⋅b₂₃,

c₃₁ = a₃₁⋅b₁₁ + a₃₂⋅b₂₁,

c₃₂ = a₃₁⋅b₁₂ + a₃₂⋅b₂₂,

c₃₃ = a₃₁⋅b₁₃ + a₃₂⋅b₂₃.

Multiplication of Matrices follows following Properties
Associative Law: (AB)C = A(BC)

Also Read Application of Matrices

2.0Transpose of Matrices

The transpose of a matrix is generated by interchanging its rows and columns matrix.. For a matrix A with dimensions m×n, the transpose A^T has dimensions n×m.

Let's consider a matrix A with elements a_ij:

$A = a_{11} a_{21} ⋮ a_{m 1} a_{12} a_{22} ⋮ a_{m 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{mn}$

The transpose AT is formed by interchanging rows and columns:

$A^{T} = a_{11} a_{12} ⋮ a_{1 n} a_{21} a_{22} ⋮ a_{2 n} \dots \dots ⋱ \dots a_{1 m} a_{m 2} ⋮ a_{mn}$

In other words, the (i, j)^th element of A^T is the (j, i)^thelement of A, i.e.,

( A^T)_ij= A_ji.

The transpose operation is important in various matrix operations and has several properties, such as:

(A^T)^T = A: Transposing a matrix twice returns the original matrix.
(kA)^T = k(A^T ): Transposing a scalar multiple of a matrix is the same as multiplying the transpose of the matrix by the scalar.
(A+B)^T =A^T + B^T: Transposing the sum of two matrices is the same as taking the sum of their transposes.
(AB)^T = B^TA^T : Transposing the product of two matrices is equivalent to multiplying their transposes in the reverse order.

Also Read Adjoint of a Matrix

3.0Symmetric and Skew Symmetric Matrices

Symmetric Matrix

A square matrix A = [a_ij] is said to be, symmetric if, a_ij =a_ij ∀ i and j(conjugate elements are equal).

Hence for symmetric matrix A = A^T.

Skew Symmetric Matrix

Square Matrix A = [a_ij] is said to be skew symmetric if, a_ij = –a_ij ∀ i and j(the pair of conjugate elements are additive inverse of each other). For a skew symmetric matrix, A = – A^T.

4.0Invertible Matrices

A square matrix is A = [aij] is said to be invertible if |A| ≠ 0, and AA^–1 = A^–1A = I.

Also Read Inverse Matrix

5.0Important Formula of Matrices

A(adj.A) = | A | I_n = (adj A) A
| adj A | = | A |^n–1 (Thus A (adj A) is always a scalar matrix)
adj (adj.A) = | A |^n–2 A, where |A| ≠ 0
|adj (adj.A)| = A, where |A| ≠ 0
adj (AB) = (adj B) (adj A)
adj (A^m) = (adj A)^m,
adj (KA) = K^n–1 (adj A), K is scalar
adj (I_n) = I_n,
adj (0) = 0
A is symmetric ⇒ adj A is also symmetric
A is diagonal ⇒ adj A is also diagonal
A is triangular ⇒ adj A is also triangular
A is singular ⇒ | adj A | = 0

6.0Solved Problems on Matrices

Example 1: If $A = 132325$ & $B = - 1 03 - 2 51$ and A+ B –D = 0 (Zero matrix), then D matrix will be ?

(A) $036275$

(B) $035276$

(C) $035176$

(D) $0 - 3 - 5 - 2 - 7 - 6$

Ans. (C)

Solution:

$L e t D = a c e b d f$

∴ $A+B-D = 132325 + - 1 03 - 2 51 - a c e b d f$

$\Rightarrow 1 - 1 - a 3 + 0 - c 2 + 3 - e 3 - 2 - b 2 + 5 - d 5 + 1 - f = 000000$

⇒ –a = 0 ⇒ a = 0, 1 –b = 0 ⇒ b = 1,

⇒ 3–c = 0 ⇒ c = 3,7 – d = 0 ⇒ d = 7,

⇒ 5 – e = 0 ⇒ e = 5,6 – f = 0 ⇒ f = 6

$∴ D = 035176$

Example 2: If A,B are two matrices such that A+B= $[1224]$ , A-B= $[3 - 2 20]$ , then find AB.

Solution:

Given A+B= $[1224]$ …(i) &

$\Rightarrow A - B = [3 - 2 20]$ …(ii) &

Adding (i) & (ii)

$\Rightarrow 2 A = [4044] \Rightarrow A = [2022]$

Subtracting (ii) from (i)

$\Rightarrow 2 B = [- 2 4 04] \Rightarrow B = [- 1 2 02]$

Now A B=

$[2022] [- 1 2 02]$

$= [2444]$

Example 3: If A= $[31 - 4 - 1]$ , show that $A^{k} = [1 + 2 k k - 4 k 1 - 2 k]$ , where k is any positive integer.

Solution: We have

$\Rightarrow A^{2} = [31 - 4 - 1] [31 - 4 - 1] = [52 - 8 - 3] = [1 + 2 \times 2 2 - 4 \times 2 1 - 2 \times 2]$

and

$\Rightarrow A^{3} = [52 - 8 - 3] [31 - 4 - 1] = [73 - 12 - 5] = [1 + 2 \times 3 3 - 4 \times 3 1 - 2 \times 3]$

Thus, it is true for indices 2 and 3. Now assume

$\Rightarrow A^{k} = [1 + 2 k k - 4 k 1 - 2 k]$

Then,

$\Rightarrow A^{k + 1} = [1 + 2 k k - 4 k 1 - 2 k] [31 - 4 - 1]$

$= [3 + 2 k k + 1 - 4 (k + 1) - 1 - 2 k]$

$\Rightarrow [1 + 2 (k + 1) k + 1 - 4 (k + 1) 1 - 2 (k + 1)]$

Example 4: If A is symmetric as well as skew symmetric matrix, then A is-

(A) diagonal matrix (B) null matrix

(C) triangular matrix (D) none of these

Ans. (B)

Solution: Let A = [a_ij] Since A is skew symmetric a_ij = –a_ji

for i = j, a_ii = –a_ii ⇒ a_ii =0

for i ≠ j, a_ij = – a_ji [∵Ais skew symmetric ] and a_ij = a_ji[∵ A is symmetric]

∴ a_ij = 0 for all i ≠ j

So, a_ij = 0 for all ‘i’ and ‘j’ i.e. A is a null matrix.

Example 5: If A= $222022002$ , then adj (adj A) is equal to

(A) $8111011001$

(B) $16111011001$

(C) $64111011001$

(D) None of these

Ans. (B)

Solution:

|A|= $222022002 = 8$

Now adj (adj A) = |A|^3–2A

= $8222022002 = 16111011001$

7.0Solved Questions on Matrices

Q1. What is an invertible matrix?

Ans: An invertible matrix, also known as a nonsingular matrix, is a square matrix that has an inverse. An invertible matrix A satisfies the condition that there exists another matrix A⁻¹ such that AA⁻¹ =A⁻¹A = I, where I represent the identity matrix.

Q2. How do you find the determinant of a matrix?

Ans: The determinant of a square matrix can be found using various methods such as cofactor expansion.

For a 2 × 2 matrix $[a c b d]$ , the determinant is given by ad−bc. For larger matrices, the process involves recursive calculations based on cofactors.

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Understand the concept and notation of matrices.
Identify different types of matrices.
Perform matrix addition, subtraction, multiplication, and scalar multiplication.
Apply the properties of transpose matrices.
Differentiate between symmetric and skew-symmetric matrices.
Perform elementary row and column operations.
Solve problems involving invertible matrices.
Apply matrix concepts in JEE Main, JEE Advanced, and CBSE Board-level questions.

2.05-Minute Quick Revision

Must-Revise Concepts

Matrix
Order of a Matrix
Row Matrix
Column Matrix
Square Matrix
Diagonal Matrix
Identity Matrix
Scalar Matrix
Zero Matrix
Symmetric Matrix
Skew-Symmetric Matrix
Transpose of a Matrix
Matrix Addition
Matrix Multiplication
Elementary Operations

Important Formula Recall

Matrix Addition:

$A + B = [a_{ij} + b_{ij}]$

Scalar Multiplication: $k A = [k a_{ij}]$
Matrix Multiplication: C=AB

where $c_{ij} = \sum_{k = 1}^{n} a_{ik} b_{kj}$

Transpose: $(A^{T}) ij = A ji$
Symmetric Matrix: $A = A^{T}$
Skew-Symmetric Matrix: $A^{T} = - A$

Important Properties

Matrix multiplication is not commutative, i.e., (AB \ne BA) in general.
Matrix multiplication is associative and distributive over addition.
The transpose of the transpose of a matrix is the original matrix.
Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
Identity matrix acts as the multiplicative identity for square matrices.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Types of Matrices	High	Easy
Matrix Operations	Very High	Moderate
Matrix Multiplication	Very High	Moderate
Transpose & Properties	High	Easy–Moderate
Symmetric & Skew-Symmetric Matrices	High	Moderate
Elementary Operations	High	Moderate

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Matrices in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Matrices is a highly scoring chapter, but students often lose marks due to incorrect matrix multiplication, order mismatches, and property-based errors. A systematic approach can help avoid these mistakes.

Common Mistake	Correct Approach
Adding or subtracting matrices of different orders	Ensure both matrices have the same order before performing addition or subtraction.
Multiplying matrices with incompatible dimensions	Verify that the number of columns of the first matrix equals the number of rows of the second matrix.
Assuming matrix multiplication is commutative	Remember that, in general, (AB \neq BA).
Confusing transpose with inverse	Transpose changes rows into columns, whereas the inverse satisfies (AA^{-1}=I).
Incorrectly identifying symmetric matrices	Check whether (A=A^T).
Forgetting properties of skew-symmetric matrices	Verify that (A^T=-A) and all diagonal elements are zero.
Arithmetic mistakes during multiplication	Compute each element carefully using row-by-column multiplication.
Ignoring the order of matrices in multiplication	Preserve the order exactly as given in the question.

6.0ALLEN Faculty Tips

Learn all types of matrices with their defining properties.
Practice row-by-column multiplication until it becomes error-free.
Memorize important transpose properties and matrix identities.
Solve conceptual questions on symmetric and skew-symmetric matrices regularly.
Verify matrix dimensions before every operation.
Revise standard matrix properties daily during exam preparation.
Practice Previous Years' Questions (PYQs) to improve speed and accuracy.

7.0PYQ Trend Analysis

Matrices is one of the highest-weightage chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Questions are frequently asked on matrix operations, transpose, symmetric matrices, elementary transformations, and conceptual properties.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Types of Matrices	1–2	Easy
Matrix Operations	2–3	Moderate
Matrix Multiplication	2–3	Moderate
Transpose & Properties	1–2	Easy–Moderate
Symmetric & Skew-Symmetric Matrices	1–2	Moderate
Elementary Operations	1–2	Moderate

8.0Most Common Question Types

Identifying different types of matrices.
Performing matrix addition, subtraction, and scalar multiplication.
Solving matrix multiplication problems.
Finding the transpose of a matrix and applying its properties.
Verifying whether a matrix is symmetric or skew-symmetric.
Expressing a square matrix as the sum of symmetric and skew-symmetric matrices.
Applying elementary row and column operations.
Solving conceptual and numerical questions involving multiple matrix properties.

9.0Smart Revision Strategy

Follow this revision sequence to master the chapter efficiently:

Revise the definitions and properties of all types of matrices.
Memorize important formulas and transpose properties.
Practice matrix addition, subtraction, and scalar multiplication.
Solve matrix multiplication questions of varying difficulty.
Revise symmetric and skew-symmetric matrix concepts.
Practice elementary transformation-based problems.
Solve Previous Years' Questions (PYQs).
End with a timed chapter-wise practice test to improve speed and accuracy.

Class 12 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

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Class 12 Maths Chapter 3: Matrices

From computer graphics and artificial intelligence to cryptography, economics, and engineering, Matrices provide a powerful mathematical framework for organizing and solving complex problems involving multiple variables. This chapter introduces the concepts of types of matrices, matrix operations, transpose, symmetric and skew-symmetric matrices, elementary operations, and invertible matrices. As one of the most application-oriented topics in Mathematics, Matrices form the backbone of Linear Algebra and play a significant role in JEE Main, JEE Advanced, and CBSE Board examinations.

At ALLEN, our expert faculty have designed these Class 12 Maths Chapter 3: Matrices Revision Notes to help students revise the chapter efficiently with complete conceptual clarity. These notes include concise theory, important formulas, matrix properties, shortcut techniques, solved illustrations, common mistakes, and PYQ-based insights. Whether you're preparing for CBSE Board exams or competitive exams like JEE Main and JEE Advanced, these revision notes will strengthen your fundamentals and improve problem-solving accuracy.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	20–25 Minutes
Key Concepts Covered	Types of Matrices, Matrix Operations, Invertible Matrices

Important topic/subtopics covered:-

Topic	Overview
Matrix	Definition and notation
Types of Matrices	Row, Column, Square, Diagonal, Identity, Zero, Scalar, Symmetric, Skew-Symmetric
Matrix Operations	Addition, subtraction, scalar multiplication, multiplication
Transpose of a Matrix	Definition and properties
Symmetric & Skew-Symmetric Matrices	Characteristics and applications
Elementary Operations	Row and column transformations
Invertible Matrices	Definition and basic properties
Applications	Linear equations, transformations, and higher mathematics

2.0Chapter 3 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Matrices

A Matrix is an organized rectangular array comprising numbers, symbols, or expressions that are systematically arranged in rows and columns. It is typically denoted by uppercase letters and is used in Mathematics and Computer science to represent data or perform operations such as addition, subtraction, multiplication, and more.

1.0Types of Matrices

Column Matrix

A matrix is termed a Column Matrix if it contains only one column.

Example: $A = 03 - 2 \frac{1}{2}$

Row Matrix

A matrix is considered a Row Matrix if it consists of only one row.

Example - A = [1 5 7]

Square Matrix

A square matrix is one where the number of rows equals the number of columns.

Example : $A = [5173]$

Diagonal Matrix

A square matrix B = [b_ij] of order m × m is classified as a Diagonal Matrix if all its non-diagonal elements are zero. In other words, a matrix B = [b_ij] of size m × m is designated as a diagonal matrix if b_ij = 0 whenever i ≠ j.

$1000050000700003$

Scalar Matrix

A Scalar Matrix is a square matrix where all diagonal elements are equal (scalar), and all non-diagonal elements are zero.

Identity Matrix

A square matrix in which ones are on the main diagonal and zeros anywhere else is known as an identity matrix. It is denoted as I_n for an n × n matrix.

$100010001$

Zero Matrix

A zero matrix, denoted as 0, is a matrix in which all elements are zero.

$000000000$

Orthogonal Matrix

An orthogonal matrix is a square matrix where the transpose is equal to its inverse.

A^T = A^–1

$cos θ - sin θ 0 - sin θ cos θ 0 001$

Also Read Types of Matrices

1.0Operation on Matrices

Addition of two Matrices

To add two matrices, you add corresponding elements in the same position in each matrix. Here is the general formula for adding two matrices:

If two matrices A = [a_ij] and B=[b_ij] have same order m×n, then their sum A + B is given by:

A + B=[a_ij] + [b_ij] = [a_ij +b_ij]

Example:

$A = [475839] B = [716223]$

$A + B = [4 + 7 7 + 1 5 + 6 8 + 2 3 + 2 9 + 3] = [1181110512]$

Addition of matrices follows the following Properties
Commutative Law: A + B = B + A
Associative Law: (A + B) + C = A + (B + C)
Additive Identity: O (Zero Matrix) is the Additive Identity for matrix addition.
Additive Inverse: –A is the additive inverse of A or negative of A.

Multiplication of a matrix by a scalar

Multiplying a matrix by a scalar involves multiplying each element of the matrix by the scalar value. For example, if we have a matrix A and a scalar k, the product kA is obtained by multiplying each element of A by k.

Example:

$A = [475839]$

$kA = k \times [475839] = [4 k 7 k 5 k 8 k 3 k 9 k]$

Scalar Multiplication of a matrix follows the following Properties
k(A +B) = k A + kB,

Multiplication of Matrices

Matrix multiplication is a binary operation that combines two matrices to produce a new matrix. For matrix multiplication to be defined the number of columns in matrix A should match the number of rows in matrix B.

Let's consider two matrices, A and B:

$A = a_{11} a_{21} a_{31} a_{12} a_{22} a_{32}$

$B = [b_{11} b_{21} b_{12} b_{22} b_{13} b_{23}]$

To perform matrix multiplication between matrices A and B, we compute each element of the resulting matrix C by taking the scalar product of the corresponding row of matrix A and the respective column of matrix B:

To multiply A and B, we follow the rule that the (i,j)^th element of the resulting matrix C is obtained by taking the scalar product of the i^th row of A and the j^th column of B:

$C = A \cdot B = c_{11} c_{21} c_{31} c_{12} c_{22} c_{32} c_{13} c_{23} c_{33}$

where

c₁₁ = a₁₁⋅b₁₁ + a₁₂⋅b₂₁,

c₁₂ = a₁₁⋅b₁₂ + a₁₂⋅b₂₂,

c₁₃ = a₁₁⋅b₁₃ + a₁₂⋅b₂₃,

c₂₁ = a₂₁⋅b₁₁ + a₂₂⋅b₂₁,

c₂₂ = a₂₁⋅b₁₂ + a₂₂⋅b₂₂,

c₂₃ = a₂₁⋅b₁₃ + a₂₂⋅b₂₃,

c₃₁ = a₃₁⋅b₁₁ + a₃₂⋅b₂₁,

c₃₂ = a₃₁⋅b₁₂ + a₃₂⋅b₂₂,

c₃₃ = a₃₁⋅b₁₃ + a₃₂⋅b₂₃.

Multiplication of Matrices follows following Properties
Associative Law: (AB)C = A(BC)

Also Read Application of Matrices

2.0Transpose of Matrices

The transpose of a matrix is generated by interchanging its rows and columns matrix.. For a matrix A with dimensions m×n, the transpose A^T has dimensions n×m.

Let's consider a matrix A with elements a_ij:

$A = a_{11} a_{21} ⋮ a_{m 1} a_{12} a_{22} ⋮ a_{m 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{mn}$

The transpose AT is formed by interchanging rows and columns:

$A^{T} = a_{11} a_{12} ⋮ a_{1 n} a_{21} a_{22} ⋮ a_{2 n} \dots \dots ⋱ \dots a_{1 m} a_{m 2} ⋮ a_{mn}$

In other words, the (i, j)^th element of A^T is the (j, i)^thelement of A, i.e.,

( A^T)_ij= A_ji.

The transpose operation is important in various matrix operations and has several properties, such as:

(A^T)^T = A: Transposing a matrix twice returns the original matrix.
(kA)^T = k(A^T ): Transposing a scalar multiple of a matrix is the same as multiplying the transpose of the matrix by the scalar.
(A+B)^T =A^T + B^T: Transposing the sum of two matrices is the same as taking the sum of their transposes.
(AB)^T = B^TA^T : Transposing the product of two matrices is equivalent to multiplying their transposes in the reverse order.

Also Read Adjoint of a Matrix

3.0Symmetric and Skew Symmetric Matrices

Symmetric Matrix

A square matrix A = [a_ij] is said to be, symmetric if, a_ij =a_ij ∀ i and j(conjugate elements are equal).

Hence for symmetric matrix A = A^T.

Skew Symmetric Matrix

Square Matrix A = [a_ij] is said to be skew symmetric if, a_ij = –a_ij ∀ i and j(the pair of conjugate elements are additive inverse of each other). For a skew symmetric matrix, A = – A^T.

4.0Invertible Matrices

A square matrix is A = [aij] is said to be invertible if |A| ≠ 0, and AA^–1 = A^–1A = I.

Also Read Inverse Matrix

5.0Important Formula of Matrices

A(adj.A) = | A | I_n = (adj A) A
| adj A | = | A |^n–1 (Thus A (adj A) is always a scalar matrix)
adj (adj.A) = | A |^n–2 A, where |A| ≠ 0
|adj (adj.A)| = A, where |A| ≠ 0
adj (AB) = (adj B) (adj A)
adj (A^m) = (adj A)^m,
adj (KA) = K^n–1 (adj A), K is scalar
adj (I_n) = I_n,
adj (0) = 0
A is symmetric ⇒ adj A is also symmetric
A is diagonal ⇒ adj A is also diagonal
A is triangular ⇒ adj A is also triangular
A is singular ⇒ | adj A | = 0

6.0Solved Problems on Matrices

Example 1: If $A = 132325$ & $B = - 1 03 - 2 51$ and A+ B –D = 0 (Zero matrix), then D matrix will be ?

(A) $036275$

(B) $035276$

(C) $035176$

(D) $0 - 3 - 5 - 2 - 7 - 6$

Ans. (C)

Solution:

$L e t D = a c e b d f$

∴ $A+B-D = 132325 + - 1 03 - 2 51 - a c e b d f$

$\Rightarrow 1 - 1 - a 3 + 0 - c 2 + 3 - e 3 - 2 - b 2 + 5 - d 5 + 1 - f = 000000$

⇒ –a = 0 ⇒ a = 0, 1 –b = 0 ⇒ b = 1,

⇒ 3–c = 0 ⇒ c = 3,7 – d = 0 ⇒ d = 7,

⇒ 5 – e = 0 ⇒ e = 5,6 – f = 0 ⇒ f = 6

$∴ D = 035176$

Example 2: If A,B are two matrices such that A+B= $[1224]$ , A-B= $[3 - 2 20]$ , then find AB.

Solution:

Given A+B= $[1224]$ …(i) &

$\Rightarrow A - B = [3 - 2 20]$ …(ii) &

Adding (i) & (ii)

$\Rightarrow 2 A = [4044] \Rightarrow A = [2022]$

Subtracting (ii) from (i)

$\Rightarrow 2 B = [- 2 4 04] \Rightarrow B = [- 1 2 02]$

Now A B=

$[2022] [- 1 2 02]$

$= [2444]$

Example 3: If A= $[31 - 4 - 1]$ , show that $A^{k} = [1 + 2 k k - 4 k 1 - 2 k]$ , where k is any positive integer.

Solution: We have

$\Rightarrow A^{2} = [31 - 4 - 1] [31 - 4 - 1] = [52 - 8 - 3] = [1 + 2 \times 2 2 - 4 \times 2 1 - 2 \times 2]$

and

$\Rightarrow A^{3} = [52 - 8 - 3] [31 - 4 - 1] = [73 - 12 - 5] = [1 + 2 \times 3 3 - 4 \times 3 1 - 2 \times 3]$

Thus, it is true for indices 2 and 3. Now assume

$\Rightarrow A^{k} = [1 + 2 k k - 4 k 1 - 2 k]$

Then,

$\Rightarrow A^{k + 1} = [1 + 2 k k - 4 k 1 - 2 k] [31 - 4 - 1]$

$= [3 + 2 k k + 1 - 4 (k + 1) - 1 - 2 k]$

$\Rightarrow [1 + 2 (k + 1) k + 1 - 4 (k + 1) 1 - 2 (k + 1)]$

Example 4: If A is symmetric as well as skew symmetric matrix, then A is-

(A) diagonal matrix (B) null matrix

(C) triangular matrix (D) none of these

Ans. (B)

Solution: Let A = [a_ij] Since A is skew symmetric a_ij = –a_ji

for i = j, a_ii = –a_ii ⇒ a_ii =0

for i ≠ j, a_ij = – a_ji [∵Ais skew symmetric ] and a_ij = a_ji[∵ A is symmetric]

∴ a_ij = 0 for all i ≠ j

So, a_ij = 0 for all ‘i’ and ‘j’ i.e. A is a null matrix.

Example 5: If A= $222022002$ , then adj (adj A) is equal to

(A) $8111011001$

(B) $16111011001$

(C) $64111011001$

(D) None of these

Ans. (B)

Solution:

|A|= $222022002 = 8$

Now adj (adj A) = |A|^3–2A

= $8222022002 = 16111011001$

7.0Solved Questions on Matrices

Q1. What is an invertible matrix?

Ans: An invertible matrix, also known as a nonsingular matrix, is a square matrix that has an inverse. An invertible matrix A satisfies the condition that there exists another matrix A⁻¹ such that AA⁻¹ =A⁻¹A = I, where I represent the identity matrix.

Q2. How do you find the determinant of a matrix?

Ans: The determinant of a square matrix can be found using various methods such as cofactor expansion.

For a 2 × 2 matrix $[a c b d]$ , the determinant is given by ad−bc. For larger matrices, the process involves recursive calculations based on cofactors.

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Understand the concept and notation of matrices.
Identify different types of matrices.
Perform matrix addition, subtraction, multiplication, and scalar multiplication.
Apply the properties of transpose matrices.
Differentiate between symmetric and skew-symmetric matrices.
Perform elementary row and column operations.
Solve problems involving invertible matrices.
Apply matrix concepts in JEE Main, JEE Advanced, and CBSE Board-level questions.

2.05-Minute Quick Revision

Must-Revise Concepts

Matrix
Order of a Matrix
Row Matrix
Column Matrix
Square Matrix
Diagonal Matrix
Identity Matrix
Scalar Matrix
Zero Matrix
Symmetric Matrix
Skew-Symmetric Matrix
Transpose of a Matrix
Matrix Addition
Matrix Multiplication
Elementary Operations

Important Formula Recall

Matrix Addition:

$A + B = [a_{ij} + b_{ij}]$

Scalar Multiplication: $k A = [k a_{ij}]$
Matrix Multiplication: C=AB

where $c_{ij} = \sum_{k = 1}^{n} a_{ik} b_{kj}$

Transpose: $(A^{T}) ij = A ji$
Symmetric Matrix: $A = A^{T}$
Skew-Symmetric Matrix: $A^{T} = - A$

Important Properties

Matrix multiplication is not commutative, i.e., (AB \ne BA) in general.
Matrix multiplication is associative and distributive over addition.
The transpose of the transpose of a matrix is the original matrix.
Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
Identity matrix acts as the multiplicative identity for square matrices.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Types of Matrices	High	Easy
Matrix Operations	Very High	Moderate
Matrix Multiplication	Very High	Moderate
Transpose & Properties	High	Easy–Moderate
Symmetric & Skew-Symmetric Matrices	High	Moderate
Elementary Operations	High	Moderate

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Matrices in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Matrices is a highly scoring chapter, but students often lose marks due to incorrect matrix multiplication, order mismatches, and property-based errors. A systematic approach can help avoid these mistakes.

Common Mistake	Correct Approach
Adding or subtracting matrices of different orders	Ensure both matrices have the same order before performing addition or subtraction.
Multiplying matrices with incompatible dimensions	Verify that the number of columns of the first matrix equals the number of rows of the second matrix.
Assuming matrix multiplication is commutative	Remember that, in general, (AB \neq BA).
Confusing transpose with inverse	Transpose changes rows into columns, whereas the inverse satisfies (AA^{-1}=I).
Incorrectly identifying symmetric matrices	Check whether (A=A^T).
Forgetting properties of skew-symmetric matrices	Verify that (A^T=-A) and all diagonal elements are zero.
Arithmetic mistakes during multiplication	Compute each element carefully using row-by-column multiplication.
Ignoring the order of matrices in multiplication	Preserve the order exactly as given in the question.

6.0ALLEN Faculty Tips

Learn all types of matrices with their defining properties.
Practice row-by-column multiplication until it becomes error-free.
Memorize important transpose properties and matrix identities.
Solve conceptual questions on symmetric and skew-symmetric matrices regularly.
Verify matrix dimensions before every operation.
Revise standard matrix properties daily during exam preparation.
Practice Previous Years' Questions (PYQs) to improve speed and accuracy.

7.0PYQ Trend Analysis

Matrices is one of the highest-weightage chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Questions are frequently asked on matrix operations, transpose, symmetric matrices, elementary transformations, and conceptual properties.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Types of Matrices	1–2	Easy
Matrix Operations	2–3	Moderate
Matrix Multiplication	2–3	Moderate
Transpose & Properties	1–2	Easy–Moderate
Symmetric & Skew-Symmetric Matrices	1–2	Moderate
Elementary Operations	1–2	Moderate

8.0Most Common Question Types

Identifying different types of matrices.
Performing matrix addition, subtraction, and scalar multiplication.
Solving matrix multiplication problems.
Finding the transpose of a matrix and applying its properties.
Verifying whether a matrix is symmetric or skew-symmetric.
Expressing a square matrix as the sum of symmetric and skew-symmetric matrices.
Applying elementary row and column operations.
Solving conceptual and numerical questions involving multiple matrix properties.

9.0Smart Revision Strategy

Follow this revision sequence to master the chapter efficiently:

Revise the definitions and properties of all types of matrices.
Memorize important formulas and transpose properties.
Practice matrix addition, subtraction, and scalar multiplication.
Solve matrix multiplication questions of varying difficulty.
Revise symmetric and skew-symmetric matrix concepts.
Practice elementary transformation-based problems.
Solve Previous Years' Questions (PYQs).
End with a timed chapter-wise practice test to improve speed and accuracy.

Frequently Asked Questions

Why are Matrices important for JEE Main and JEE Advanced?

Which topics from Matrices are most frequently asked in JEE?

What is the most common mistake students make in Matrix Multiplication?

How can I revise the Matrices chapter quickly before the exam?

Which chapter should I study after completing Matrices?

Join ALLEN!

Class 12 Maths Chapter 3: Matrices

1.0Chapter Snapshot

2.0Chapter 3 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Matrices

1.0Types of Matrices

Column Matrix

Row Matrix

Square Matrix

Diagonal Matrix

Scalar Matrix

Identity Matrix

Zero Matrix

Orthogonal Matrix

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page

Frequently Asked Questions

Why are Matrices important for JEE Main and JEE Advanced?

Which topics from Matrices are most frequently asked in JEE?

What is the most common mistake students make in Matrix Multiplication?

How can I revise the Matrices chapter quickly before the exam?

Which chapter should I study after completing Matrices?

Join ALLEN!

Class 12 Maths Chapter 3: Matrices

1.0Chapter Snapshot

2.0Chapter 3 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Matrices

1.0Types of Matrices

Column Matrix

Row Matrix

Square Matrix

Diagonal Matrix

Scalar Matrix

Identity Matrix

Zero Matrix

Orthogonal Matrix

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page