NCERT Class 12 Matrices: Types, Formulas, and Examples

Numbers are everywhere, from social media algorithms to computer graphics and even artificial intelligence. Behind many of these systems lies a powerful mathematical tool called matrices. In Class 12 Mathematics, matrices help students organize data, perform calculations efficiently, and understand advanced mathematical concepts. 

This chapter introduces different types of matrices, operations, and formulas used in problem-solving. In this guide, we will break down the NCERT Matrices chapter in a simple way with clear explanations, solved examples, formulas, and practice questions to help students master the topic confidently. 

What is a Matrix?

A matrix is a rectangular arrangement of numbers, symbols, or functions placed in rows and columns. Matrices help organize large amounts of data in a structured form, making mathematical calculations easier and more systematic. They are widely used in fields such as computer graphics, economics, statistics, and engineering.

Representation of a Matrix

A matrix is usually represented by capital letters such as A, B, or C.

Mathematically, a matrix is written as:

Here:

• A represents the name of the matrix.
• aᵢⱼ represents the element of the matrix.
• i represents the row number.
• j represents the column number. 

This notation means that each element in the matrix can be identified by its position (row, column).

Example of a Matrix

For example, consider the following matrix:

Understanding the Parts of a Matrix

Let us break down the matrix to understand its components.

1. Elements

The numbers inside the matrix are called elements or entries.
In the above matrix, the elements are 2, 3, 4, and 5.

2. Rows

Rows are the horizontal arrangements of elements.
In the example matrix:

• Row 1 → 2, 3
• Row 2 → 4, 5 

So, the matrix has 2 rows.

3. Columns

Columns are the vertical arrangements of elements.

• Column 1 → 2, 4
• Column 2 → 3, 5 

So, the matrix has 2 columns.

Order of a Matrix

The order of a matrix is written as:

For the above example:

• Rows = 2
• Columns = 2 

So, the order of the matrix is 2 × 2.

Understanding matrices and their structure is the first step toward learning matrix operations and advanced applications in mathematics. 

Order of a Matrix

The order of a matrix tells us the size or dimension of the matrix. It shows how many rows and columns are present in the matrix.

Definition

If a matrix has m rows and n columns, then the order of the matrix is written as:

Here:

• m represents the number of rows
• n represents the number of columns 

So, a matrix with 2 rows and 3 columns will have the order 2 × 3.

Example

Consider the matrix:

Let us identify the rows and columns:

• Rows = 2
• Columns = 3 

Therefore, the order of matrix A = 2 × 3.

Important Points to Remember

Total number of elements

The total number of elements in a matrix is:

For example, in a 2 × 3 matrix, the total number of elements is 2 × 3 = 6.

Position of an element

Each element in a matrix is represented as aᵢⱼ, where:

• i represents the row number
• j represents the column number 

For example:

• a₁₂ means the element in row 1 and column 2. 

Example Problem

Find the order of the matrix:

Solution

Step 1: Count the number of rows
Rows = 3

Step 2: Count the number of columns
Columns = 2

Step 3: Write the order

Order of matrix A = 3 × 2

So, this matrix has 3 rows and 2 columns, which means it is a 3 × 2 matrix. 

Types of Matrices

Matrices can be classified into different types based on their structure and elements. Understanding these types helps students identify matrices easily and apply the correct formulas while solving problems.

Row Matrix

A row matrix is a matrix that has only one row but can have multiple columns.

This means the order of a row matrix is always 1 × n.

Example

Order: 1 × 3

Here, there is one row and three columns, so it is a row matrix.

Column Matrix

A column matrix is a matrix that has only one column but can have multiple rows.

The order of a column matrix is always m × 1.

Example

Order: 3 × 1

This matrix has three rows and one column, so it is called a column matrix.

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Square Matrix

A square matrix is a matrix in which the number of rows is equal to the number of columns.

If a matrix has n rows and n columns, its order is n × n.

Example

Order: 2 × 2

Since the number of rows and columns are equal, it is a square matrix.

Zero Matrix

A zero matrix (or null matrix) is a matrix in which all the elements are equal to zero.

Example

No matter what the order is, if every element is 0, the matrix is called a zero matrix.

Diagonal Matrix

A diagonal matrix is a square matrix in which all the elements except the main diagonal are zero.

The main diagonal runs from the top-left corner to the bottom-right corner.

Example

Here, only the diagonal elements (3, 5, 2) are non-zero, and all other elements are 0.

Scalar Matrix

A scalar matrix is a special type of diagonal matrix in which all the diagonal elements are equal.

Example

Here, the diagonal elements are 4 and 4, which are equal. Therefore, it is called a scalar matrix.

Identity Matrix

An identity matrix (also called a unit matrix) is a special type of scalar matrix in which all diagonal elements are 1 and all other elements are 0.

Example

This matrix is called the identity matrix because when it is multiplied with another matrix, the result remains the same matrix.

Understanding these different types of matrices helps students quickly recognize matrix structures and apply the correct operations while solving mathematical problems. 

Equality of Matrices

Two matrices are said to be equal when they satisfy certain conditions. Matrix equality is important because it helps us find unknown values in matrices and compare two matrices.

Conditions for Equality of Matrices

Two matrices A and B are equal if:

1. They have the same order (same number of rows and columns).  
2. All corresponding elements are equal. 

This means that the element in the same position in both matrices must be equal.

Mathematically, if

then the matrices are equal only when

for every value of i and j.

Example Problem

If

and

Find the values of x and y.

Solution

Step 1: Compare the elements in the same positions.

First row, first column:

First row, second column:

Second row, first column:

Second row, second column:

Final Answer

So, when x = 4 and y = 5, the matrices A and B become equal. 

Addition of Matrices

Matrix addition is one of the most basic operations in matrix algebra. It involves adding the corresponding elements of two matrices.

Rule for Matrix Addition

Two matrices can be added only when they have the same order.

This means both matrices must have the same number of rows and the same number of columns.

If the orders are different, matrix addition is not possible.

Formula for Addition of Matrices

If two matrices A = [aᵢⱼ] and B = [bᵢⱼ] have the same order, then their sum is obtained by adding the corresponding elements.

This means each element in the resulting matrix is the sum of the corresponding elements of matrices A and B.

Example

Let

and

Both matrices have the same order (2 × 2), so they can be added.

Solution

Add the corresponding elements:

First row:
1 + 5 = 6
2 + 6 = 8

Second row:
3 + 7 = 10
4 + 8 = 12

Result:

So, the sum of matrices A and B is:

Matrix addition helps simplify calculations and is widely used in data processing, engineering, and computer science. 

Scalar Multiplication of a Matrix

Scalar multiplication means multiplying every element of a matrix by a single number. The number used to multiply the matrix is called a scalar.

Definition

If k is a number (scalar) and A is a matrix, then multiplying the matrix by k means multiplying each element of the matrix by k.

Here:

• k represents the scalar number.
• aᵢⱼ represents the elements of the matrix.
• Each element in matrix A is multiplied by k. 

Example

Multiply the matrix by 2.

Solution

Multiply 2 with every element of the matrix.

First row:
2 × 1 = 2
2 × 3 = 6

Second row:
2 × 4 = 8
2 × 2 = 4

So the resulting matrix becomes:

Therefore,

Scalar multiplication is useful when scaling values in data tables, physics calculations, and mathematical modeling. 

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Negative of a Matrix

The negative of a matrix is obtained by multiplying every element of the matrix by −1. This operation changes the sign of each element in the matrix.

Definition

If A is a matrix, then its negative is written as:

This means that every element in matrix A is multiplied by −1.

Example

Let the matrix be:

To find −A, multiply each element by −1.

Solution

Multiply −1 with each element:

First row:
−1 × 2 = −2
−1 × 3 = −3

Second row:
−1 × 4 = −4
−1 × 5 = −5

So the negative matrix becomes:

Therefore, the negative of matrix A is:

The concept of the negative of a matrix is useful in matrix operations like matrix subtraction and solving algebraic equations involving matrices. 

Multiplication of Matrices

Matrix multiplication is another important operation in matrix algebra. Unlike addition and subtraction, matrix multiplication follows a specific condition. It is widely used in computer graphics, data analysis, and engineering calculations.

Condition for Matrix Multiplication

Two matrices A and B can be multiplied only when:

Number of columns of A = Number of rows of B

If this condition is satisfied, the product AB exists.

For example:

• If A is of order m × n
• And B is of order n × p 

Then the product AB will be of order m × p.

Formula for Matrix Multiplication

If A and B are two matrices, their product is written as:

Each element of matrix C is obtained by multiplying the elements of the corresponding row of A with the column of B and then adding them.

This formula shows how each element of the resulting matrix is calculated.

Example (Step-by-Step)

Multiply the matrices:

Both matrices are 2 × 2, so multiplication is possible.

Step 1: Find the first element

First row of A × First column of B

Step 2: Find the second element

First row of A × Second column of B

Step 3: Find the third element

Second row of A × First column of B

Step 4: Find the fourth element

Second row of A × Second column of B

Final Answer

So, the product of matrices A and B is:

Matrix multiplication is an essential concept that forms the foundation for advanced topics like determinants, inverse matrices, and linear algebra applications. 

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Properties of Matrix Multiplication

Matrix multiplication follows certain mathematical properties. Understanding these properties helps students solve problems faster and recognize how matrices behave during operations.

Non-Commutative Property

In matrix multiplication, the order of multiplication matters.

This means:

So, multiplying matrix A × B does not always give the same result as B × A. In some cases, BA may not even be possible if the order condition is not satisfied.

Example

Let

First find AB:

Now find BA:

Since

matrix multiplication is non-commutative.

Associative Property

Matrix multiplication follows the associative property, which means the grouping of matrices does not affect the result.

This means if three matrices A, B, and C are multiplied, we can first multiply A and B or B and C — the final result will remain the same.

Example

Let

First compute:

Now compute:

Since both results are equal, the associative property holds true.

Distributive Property

Matrix multiplication also follows the distributive property over addition.

This means multiplication distributes over matrix addition.

and

Example

Let

First calculate B + C:

Now multiply:

Now calculate the right side:

Since both sides give the same result, the distributive property is verified.

Key Takeaway

The three important properties of matrix multiplication are:

• Non-commutative:
• Associative:
• Distributive:

Understanding these properties helps students simplify complex matrix calculations and solve exam problems more efficiently. 

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Transpose of a Matrix

The transpose of a matrix is obtained by interchanging its rows and columns. In simple terms, the rows of the original matrix become columns, and the columns become rows in the new matrix.

The transpose of a matrix A is written as Aᵀ.

Definition

If a matrix A = [aᵢⱼ], then its transpose is obtained by switching the position of the indices.

This means the element in the i-th row and j-th column of matrix A becomes the element in the j-th row and i-th column in the transpose.

Example

Let the matrix be:

To find the transpose, we interchange rows and columns.

• First row (1, 2) becomes the first column.
• Second row (3, 4) becomes the second column. 

So, the transpose of matrix A is:

Properties of Transpose

Transpose follows several important mathematical properties.

1. Transpose of Transpose

If we take the transpose of a matrix and then transpose it again, we get the original matrix.

This means applying transpose twice returns the same matrix.

2. Transpose of a Sum

The transpose of the sum of two matrices is equal to the sum of their transposes.

This property works only when the matrices have the same order.

3. Transpose of a Product

The transpose of the product of two matrices is equal to the product of their transposes in reverse order.

Notice that the order of multiplication is reversed.

Key Takeaway

The transpose operation helps simplify matrix expressions and is widely used in linear algebra, matrix equations, and advanced mathematical computations. Understanding these properties makes it easier for students to solve matrix-related problems in Class 12 mathematics. 

Symmetric Matrix

A symmetric matrix is a special type of square matrix in which the matrix is equal to its transpose. This means that when the rows and columns are interchanged, the matrix remains the same.

Condition for a Symmetric Matrix

A matrix A is symmetric if:

This means the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column.

In simple words, the matrix is mirror-like along its main diagonal.

Example

Consider the matrix:

Now find its transpose.

Since

the matrix A is a symmetric matrix.

Key Point

In a symmetric matrix:

• The matrix must always be a square matrix.
• Elements on opposite sides of the main diagonal are equal. 

For example:

Symmetric matrices are important in advanced mathematics, physics, statistics, and computer science because they simplify many calculations. 

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Skew-Symmetric Matrix

A skew-symmetric matrix is a special type of square matrix in which the transpose of the matrix is equal to the negative of the matrix.

In simple terms, when we take the transpose of the matrix, the elements become the negative values of the corresponding elements in the original matrix.

Condition for a Skew-Symmetric Matrix

A matrix A is skew-symmetric if:

This means that the element in the i-th row and j-th column is the negative of the element in the j-th row and i-th column.

For example:

Example

Consider the matrix:

Now find its transpose.

If we take the negative of matrix A:

Since

the matrix A is a skew-symmetric matrix.

Important Rule

In a skew-symmetric matrix:

All diagonal elements are always equal to 0.

This happens because:

which is possible only when

Key Takeaway

• A skew-symmetric matrix must be a square matrix.
• Opposite elements have opposite signs.
• Diagonal elements are always zero. 

Skew-symmetric matrices are useful in advanced algebra, physics, and linear transformations.

Invertible Matrix

An invertible matrix is a special type of square matrix that has another matrix called its inverse. When a matrix is multiplied by its inverse, the result is the identity matrix.

Definition

A square matrix A is said to be invertible if there exists a matrix A⁻¹ such that:

Here:

• A = Original matrix
• A⁻¹ = Inverse of matrix A
• I = Identity matrix 

This means multiplying a matrix by its inverse always gives the identity matrix.

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What is an Inverse Matrix?

The inverse of a matrix is another matrix that reverses the effect of the original matrix when multiplied.

If a matrix A has an inverse A⁻¹, then:

However, not every matrix has an inverse. A matrix must satisfy certain conditions:

• It must be a square matrix.
• Its determinant must not be zero.  

If the determinant is zero, the matrix is called a singular matrix and it does not have an inverse.

Identity Matrix Relation

The identity matrix plays an important role in matrix algebra.

An identity matrix is a square matrix in which:

• All diagonal elements are 1
• All other elements are 0 

Example of a 2 × 2 identity matrix:

The identity matrix works like the number 1 in multiplication.

For any matrix A:

This means multiplying any matrix by the identity matrix does not change the matrix.

Key Idea

If a matrix A has an inverse A⁻¹, then:

This concept is important for solving systems of linear equations, matrix algebra, and advanced topics in linear algebra. 

Inverse of a Matrix Using Elementary Operations

The inverse of a matrix can be found using elementary row operations. This method is commonly used in Class 12 mathematics because it provides a systematic way to convert a matrix into its inverse.

The basic idea is to transform the given matrix into the identity matrix. During this process, the identity matrix on the right side transforms into the inverse of the matrix.

Steps to Find the Inverse Using Elementary Operations

Step 1: Write the matrix in the form

Practically, we write the matrix along with the identity matrix as an augmented matrix:

Step 2: Apply elementary row operations

Use operations such as:

• Interchanging two rows
• Multiplying a row by a non-zero number
• Adding or subtracting rows 

These operations help simplify the matrix.

Step 3: Convert the matrix into the identity matrix

Apply row operations until the left side becomes the identity matrix.

Step 4: The right-hand side becomes the inverse

Once the left side becomes I, the right side automatically becomes .

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Differential Equations Made Easy for Class 12 Students

Example

Find the inverse of the matrix

Step 1: Write the augmented matrix

Step 2: Make the first column below the pivot zero

Perform:

Step 3: Make the pivot element equal to 1

Divide the second row by −2

Step 4: Make the element above the pivot zero

Perform:

Final Result

Now the left side has become the identity matrix, so the right side is the inverse of A.

Key Idea

Using elementary row operations, we transform:

This method is very useful for finding the inverse of matrices in exams and solving systems of equations. 

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Solved Examples from NCERT

To understand matrices better, it is important to practice NCERT-style problems. Below are some commonly asked types of questions from the Matrices chapter along with clear step-by-step solutions.

Example 1: Construct a Matrix Using

Construct a 3 × 3 matrix whose elements are given by:

Step 1: Identify the order

The matrix is 3 × 3, so:

Step 2: Find each element

Final Matrix

Example 2: Check Which Matrices Can Be Added

Let

Step 1: Identify the order

• Matrix A = 2 × 2
• Matrix B = 2 × 2
• Matrix C = 1 × 3 

Step 2: Apply matrix addition rule

Matrices can be added only if their orders are the same.

Step 3: Check combinations

• A + B → Possible (same order)
• A + C → Not possible
• B + C → Not possible 

Step 4: Perform addition

Example 3: Prove Symmetric + Skew-Symmetric Matrix = Zero Matrix

Let A be a symmetric matrix and B be a skew-symmetric matrix.

For symmetric matrix:

For skew-symmetric matrix:

Suppose:

where O is the zero matrix.

Step 1: Take transpose

Using transpose property:

Step 2: Substitute properties

Since ,

Therefore,

Hence, the sum of a symmetric and skew-symmetric matrix under this condition becomes a zero matrix.

Example 4: Find the Value of

Given:

If A = B, find x.

Step 1: Apply equality rule

For two matrices to be equal:

• Their orders must be the same
• Their corresponding elements must be equal 

Step 2: Compare elements

First element:

Other elements are already equal.

Final Answer

Example 5: Find the Transpose of a Matrix

Given:

Step 1: Identify order

Matrix A is 2 × 3.

Step 2: Interchange rows and columns

First row → First column
Second row → Second column

Final Transpose

Order becomes 3 × 2.

Example 6: Matrix Multiplication

Multiply:

Step 1: Multiply rows and columns

First row × first column:

First row × second column:

Second row × first column:

Second row × second column:

Final Answer

These NCERT-style solved examples help students understand how to apply matrix formulas and concepts step by step. Regular practice of such problems makes the Matrices chapter much easier and more scoring in Class 12 exams. 

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Practice Questions 

1. Basic Practice Questions

1. Find the order of the matrix  

2. Write the order of a matrix having 5 rows and 2 columns.
3. Identify the type of matrix:  

4. Identify whether the matrix is row matrix or column matrix: 

5. Find the transpose of the matrix 

6. Find the transpose of 

7. Identify whether the matrix is square matrix or rectangular matrix: 

8. Write the zero matrix of order .
9. Determine whether the matrix is diagonal matrix: 

10. Find the transpose of 

2. Intermediate Practice Questions

1. Add the matrices 

2. Subtract the matrices 

3. Multiply the matrices 

4. Find if 

5. If 

and

find such that .

6. Find if 

7. Solve the matrix equation 

where

8. If 

find .

9. Verify whether for the matrices 

10. Multiply 

with

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3. NCERT Level Questions

1. Verify that 

for suitable matrices.

2. Show that 

is symmetric matrix.

3. Prove that 

is skew symmetric matrix.

4. Verify the associative property of matrix multiplication.
5. Verify the distributive property

6. Show that the transpose of a symmetric matrix is itself.
7. If is symmetric and is skew symmetric, prove that 

is neither symmetric nor skew symmetric in general.

8. If , prove that all diagonal elements are zero.
9. Prove that 

10. Verify the property 

4. Challenge Questions

1. Express the matrix 

as the sum of symmetric and skew symmetric matrices.

2. Find the inverse of the matrix

3. Solve the matrix equation 

where

4. Show that if is symmetric and is skew symmetric then 

is skew symmetric.

5. Find the inverse using elementary row operations.
6. If , show that .
7. Prove that the inverse of a symmetric matrix is also symmetric.
8. If , prove that .
9. If is invertible, show that 

10. Solve the matrix equation 

for .

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Common Mistakes Students Make in Matrices

Students often make small mistakes while solving matrix problems. Being aware of them can help avoid errors.

Ignoring matrix order in multiplication – Matrix multiplication is only possible when the number of columns of the first matrix equals the number of rows of the second matrix.

Confusing transpose with inverse – Transpose only swaps rows and columns, while the inverse is a completely different concept.

Forgetting that – Matrix multiplication is not commutative, so changing the order changes the result.

Errors in element positions – Students sometimes place elements in the wrong position while forming matrices or calculating results.

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Confidence building – Regular practice and feedback help students overcome math anxiety and improve accuracy.

Communication + logical thinking – Students learn how to explain solutions clearly, improving both reasoning and communication skills.

Understanding the Importance of Matrices

Matrices are an important topic in Class 12 mathematics that introduces students to structured numerical systems used in many real-world applications. Through matrices, students learn concepts like matrix types, operations, transpose, and inverse. Understanding these ideas requires strong conceptual clarity and regular practice.

 Matrices are widely used in technology, science, engineering, and data analysis. With consistent practice and the right guidance, students can solve matrix problems efficiently. Structured learning platforms like PlanetSpark can make mastering matrices easier and more engaging for students.

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