NCERT Solutions for Class 12 Maths Chapter 3: Matrices

Matrices are an essential concept in higher mathematics, providing a structured and compact way of representing systems of linear equations and various transformations. Chapter 3 of the NCERT Class 12 Mathematics curriculum delves deep into matrices, their types, operations, and applications. In this article, we will explore the key concepts covered in this chapter, along with NCERT solutions for Class 12 Maths Chapter 3: Matrices.

Table of Contents

1. Introduction to Matrices
2. Types of Matrices
3. Matrix Operations
   3.1 Addition of Matrices
   3.2 Scalar Multiplication of Matrices
   3.3 Matrix Multiplication
4. Transpose of a Matrix
5. Symmetric and Skew-Symmetric Matrices
6. Determinants of a Matrix
7. NCERT Solutions for Class 12 Maths Chapter 3
   7.1 Exercise 3.1
   7.2 Exercise 3.2
   7.3 Exercise 3.3
   7.4 Miscellaneous Exercises
8. Applications of Matrices
9. FAQs on NCERT Solutions for Chapter 3 Matrices

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1. Introduction to Matrices

A matrix is a rectangular array of numbers or functions arranged in rows and columns. Matrices are used to represent linear transformations, solve systems of equations, and model various problems in physics, engineering, economics, and computer science. Mathematically, a matrix is denoted as:

$$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$

In this notation, $a_{ij}$ represents the element in the $i$-th row and $j$-th column of the matrix.

Matrix Order

The order of a matrix refers to the number of rows and columns it has. For example, a matrix with 3 rows and 3 columns is called a 3x3 matrix.

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2. Types of Matrices

Matrices can be classified into several types based on their structure and the properties of their elements. Understanding these types is crucial for solving problems efficiently.

2.1 Row Matrix

A matrix with only one row and multiple columns is called a row matrix. For example:

$$A = \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}$$

2.2 Column Matrix

A matrix with only one column and multiple rows is called a column matrix. For example:

$$B = \begin{pmatrix} 1 \\ 3 \\ 5 \end{pmatrix}$$

2.3 Square Matrix

A square matrix has an equal number of rows and columns. For example, a 2x2 square matrix is represented as:

$$C = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$$

2.4 Diagonal Matrix

A diagonal matrix is a square matrix in which all the elements outside the main diagonal are zero. The main diagonal elements can be non-zero. For example:

$$D = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{pmatrix}$$

2.5 Identity Matrix

An identity matrix is a special diagonal matrix where all the diagonal elements are 1, and all other elements are 0. It is denoted by $I_n$, where $n$ represents the order of the matrix. For example:

$$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

2.6 Zero Matrix

A zero matrix is a matrix where all the elements are zero. For example:

$$O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$$

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3. Matrix Operations

Matrix operations are fundamental for performing calculations involving matrices. The key operations are addition, scalar multiplication, and matrix multiplication.

3.1 Addition of Matrices

Two matrices can be added if they have the same order. The sum of two matrices $A$ and $B$ is another matrix $C$, where each element of $C$ is the sum of the corresponding elements of $A$ and $B$.

$$C = A + B \quad \text{where} \quad C_{ij} = A_{ij} + B_{ij}$$

For example, if

$$A = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$

Then,

$$C = \begin{pmatrix} 2+1 & 3+2 \\ 4+3 & 5+4 \end{pmatrix} = \begin{pmatrix} 3 & 5 \\ 7 & 9 \end{pmatrix}$$

3.2 Scalar Multiplication of Matrices

Scalar multiplication involves multiplying every element of a matrix by a constant value (called a scalar). If $k$ is a scalar and $A$ is a matrix, then the scalar multiplication of $A$ by $k$ is given by:

$$kA = \begin{pmatrix} k \cdot A_{11} & k \cdot A_{12} \\ k \cdot A_{21} & k \cdot A_{22} \end{pmatrix}$$

For example, if

$$A = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \quad \text{and} \quad k = 2$$

Then,

$$kA = \begin{pmatrix} 2 \cdot 2 & 2 \cdot 3 \\ 2 \cdot 4 & 2 \cdot 5 \end{pmatrix} = \begin{pmatrix} 4 & 6 \\ 8 & 10 \end{pmatrix}$$

3.3 Matrix Multiplication

Matrix multiplication involves multiplying two matrices to produce a third matrix. However, matrix multiplication is only possible when the number of columns in the first matrix is equal to the number of rows in the second matrix.

If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then their product $C = A \times B$ is an $m \times p$ matrix.

For example, if

$$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$$

Then,

$$C = A \times B = \begin{pmatrix} (1 \cdot 2 + 2 \cdot 1) & (1 \cdot 0 + 2 \cdot 3) \\ (3 \cdot 2 + 4 \cdot 1) & (3 \cdot 0 + 4 \cdot 3) \end{pmatrix} = \begin{pmatrix} 4 & 6 \\ 10 & 12 \end{pmatrix}$$

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4. Transpose of a Matrix

The transpose of a matrix $A$, denoted by $A^T$, is obtained by interchanging the rows and columns of $A$. For example, if

$$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$

Then, the transpose of $A$ is:

$$A^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$$

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5. Symmetric and Skew-Symmetric Matrices

A square matrix $A$ is called symmetric if $A = A^T$, and it is called skew-symmetric if $A = -A^T$. For example, the matrix

$$A = \begin{pmatrix} 2 & 3 \\ 3 & 2 \end{pmatrix}$$

is symmetric because its transpose is equal to the original matrix. On the other hand, the matrix

$$B = \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$$

is skew-symmetric because its transpose is the negative of the original matrix.

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6. Determinants of a Matrix

The determinant of a square matrix is a scalar value that provides important information about the matrix, such as whether it is invertible. The determinant of a 2x2 matrix

$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

is calculated as:

$$\text{det}(A) = ad - bc$$

For example, the determinant of the matrix

$$A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$$

is:

$$\text{det}(A) = (2 \cdot 4) - (3 \cdot 1) = 8 - 3 = 5$$

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7. NCERT Solutions for Class 12 Maths Chapter 3

7.1 Exercise 3.1

Exercise 3.1 focuses on the basic operations on matrices such as addition, subtraction, and scalar multiplication. Students are required to perform these operations on given matrices and simplify the results.

7.2 Exercise 3.2

Exercise 3.2 deals with matrix multiplication. Students need to multiply matrices and solve related problems. This exercise is essential for understanding matrix multiplication and its applications.

7.3 Exercise 3.3

Exercise 3.3 introduces the concept of the transpose of a matrix, symmetric matrices, and skew-symmetric matrices. Students are required to find the transpose of matrices and verify whether they are symmetric or skew-symmetric.

7.4 Miscellaneous Exercises

The miscellaneous exercises at the end of the chapter provide a comprehensive set of problems that cover all the concepts discussed in the chapter. These problems are designed to test the student's understanding and application of matrices in various contexts.

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8. Applications of Matrices

Matrices have numerous applications in various fields, including:

• Physics: Matrices are used in quantum mechanics to describe the behavior of particles and systems.
• Computer Graphics: Matrices are used to perform transformations such as translation, rotation, and scaling of images.
• Economics: Matrices are used to model economic systems and optimize resource allocation.
• Engineering: Matrices are used in control systems, electrical networks, and structural analysis.

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9. FAQs on NCERT Solutions for Chapter 3 Matrices

Q1. What is a matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. It is used to represent systems of linear equations and transformations.

Q2. What are the types of matrices?
There are various types of matrices, including row matrix, column matrix, square matrix, diagonal matrix, identity matrix, and zero matrix.

Q3. How is matrix multiplication performed?
Matrix multiplication is performed by multiplying the corresponding rows of the first matrix by the columns of the second matrix and summing the products.

Q4. What is the transpose of a matrix?
The transpose of a matrix is obtained by interchanging its rows and columns.

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In conclusion, Chapter 3 of NCERT Class 12 Maths on Matrices is crucial for students aspiring to excel in mathematics and its applications. Understanding the types of matrices, operations on matrices, and their applications is essential for solving problems efficiently. The NCERT solutions provided here serve as a guide for students to navigate through the chapter and strengthen their foundation in matrices.