CBSE Class 12 Mathematics NCERT Solutions: Chapter 3 - Matrices

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NCERT Solutions PDF Class 12 PDF

CBSE Class 12 Mathematics chapter on Matrices introduces this essential topic. These NCERT Solutions offer a thorough exploration of matrices, covering their definition, order, and the count of elements. You'll learn to pinpoint specific elements and figure out the potential orders for matrices with a set number of elements. The solutions also walk you through building matrices when the elements are defined by specific formulas. This step-by-step method ensures you understand clearly and accurately, making it a great help for your board exams. Practicing with these solutions will strengthen your understanding of matrix operations and characteristics, improving your ability to solve problems and boosting your exam confidence.

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Quick info

BoardCBSE
ClassClass 12
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterMatrices

Chapter summary

Chapter 3, Matrices, in the NCERT Solutions for Class 12 Mathematics, covers the basics of matrices. It explains how to determine the order and number of elements in a matrix, and how to identify individual elements using their position indices. The chapter also delves into finding all possible orders for a matrix given a specific number of elements and constructing matrices based on provided general element formulas. These solutions are designed to build a strong foundation in matrix theory.

Learning outcomes

  • Understand the definition and order of a matrix.
  • Determine the number of elements in a matrix.
  • Identify specific elements of a matrix using their indices.
  • Find all possible orders for a matrix with a given number of elements.
  • Construct matrices based on given element formulas.

Topics covered

Paper topics

  • Introduction to Matrices
  • Order of a Matrix
  • Number of Elements in a Matrix
  • Identifying Matrix Elements
  • Possible Orders of a Matrix
  • Construction of Matrices

Important topics

  • Order of a Matrix
  • Number of Elements
  • Identifying Elements (a_ij)
  • Possible Orders for a Given Number of Elements
  • Constructing Matrices from Formulas

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Questions and Solutions

Question 1

Consider the matrix:

A = \begin{bmatrix} 5 \ 35 \ \sqrt{3} \\ 12 \end{bmatrix}

Write:

  1. The order of the matrix.
  2. The number of elements in the matrix.
  3. Write the elements a_{13}, a_{21}, a_{33}, a_{24}, a_{23}.
Solution:

The given matrix is:

A = \begin{bmatrix} 5 & ? & ? & ? \\ 35 & ? & ? & ? \\ \sqrt{3} & ? & ? & 12 \end{bmatrix}

From the visual representation, it appears the matrix provided in the question might be incomplete or presented in a way that doesn't fully match the standard notation for a 3x4 matrix. However, based on the provided solution elements and the question asking for a 3x4 matrix, we will assume the intended matrix structure allows for these elements.

i) The order of the matrix:

To determine the order, we count the number of rows and columns. Let's assume the matrix is intended to be of order 3 \times 4 as implied by the question parts.

Number of rows = 3

Number of columns = 4

Therefore, the order of the matrix is 3 \times 4.

ii) The number of elements:

The total number of elements in a matrix is the product of its order (number of rows × number of columns).

Number of elements = 3 \times 4 = 12.

iii) Writing the specific elements:

The notation a_{ij} refers to the element in the i-th row and j-th column. Based on the elements provided in the source answer, we infer the positions:

  • a_{13}: Element in the 1st row, 3rd column. (Value inferred from source: 19)
  • a_{21}: Element in the 2nd row, 1st column. (Value: 35)
  • a_{33}: Element in the 3rd row, 3rd column. (Value inferred from source: -5)
  • a_{24}: Element in the 2nd row, 4th column. (Value inferred from source: 12)
  • a_{23}: Element in the 2nd row, 3rd column. (Value inferred from source: 2)

Note: The initial matrix representation A = \begin{bmatrix} 5 \ 35 \ \sqrt{3} \\ 12 \end{bmatrix} seems to be a column matrix or incorrectly formatted. The solution's specific elements suggest a different intended matrix structure, possibly A = \begin{bmatrix} ? & ? & 19 & ? \\ 35 & 2 & ? & 12 \\ ? & ? & -5 & ? \end{bmatrix} if we were to place the given values.

Question 2

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Solution:

The number of elements in a matrix is the product of its number of rows (m) and the number of columns (n), i.e., m \times n.

Case 1: Matrix with 24 elements

We need to find all pairs of natural numbers (m, n) such that m \times n = 24. These pairs represent the possible orders of the matrix.

The factor pairs of 24 are:

  • 1 and 24
  • 2 and 12
  • 3 and 8
  • 4 and 6

Considering both m \times n and n \times m as distinct orders, the possible orders are:

  • 1 \times 24
  • 24 \times 1
  • 2 \times 12
  • 12 \times 2
  • 3 \times 8
  • 8 \times 3
  • 4 \times 6
  • 6 \times 4

Case 2: Matrix with 13 elements

We need to find pairs of natural numbers (m, n) such that m \times n = 13.

Since 13 is a prime number, its only natural number factors are 1 and 13.

The possible ordered pairs are:

  • 1 \times 13
  • 13 \times 1

Therefore, the possible orders for a matrix with 13 elements are 1 \times 13 and 13 \times 1.

Question 3

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Solution:

The number of elements in a matrix is the product of its number of rows (m) and the number of columns (n), i.e., m \times n.

Case 1: Matrix with 18 elements

We need to find all pairs of natural numbers (m, n) such that m \times n = 18. These pairs represent the possible orders of the matrix.

The factor pairs of 18 are:

  • 1 and 18
  • 2 and 9
  • 3 and 6

Considering both m \times n and n \times m as distinct orders, the possible orders are:

  • 1 \times 18
  • 18 \times 1
  • 2 \times 9
  • 9 \times 2
  • 3 \times 6
  • 6 \times 3

Case 2: Matrix with 5 elements

We need to find pairs of natural numbers (m, n) such that m \times n = 5.

Since 5 is a prime number, its only natural number factors are 1 and 5.

The possible ordered pairs are:

  • 1 \times 5
  • 5 \times 1

Therefore, the possible orders for a matrix with 5 elements are 1 \times 5 and 5 \times 1.

Question 5

Construct a 3 \times 4 matrix, whose elements are given by the formula:

(i) a_{ij} = \frac{1}{2} |-3i + j|

(ii) a_{ij} = 2i - j

Solution:

A general 3 \times 4 matrix has the form:

A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ \end{bmatrix}

We will construct the matrix using the given formulas for the elements a_{ij}, where i represents the row number (i = 1, 2, 3) and j represents the column number (j = 1, 2, 3, 4).

Part (i): a_{ij} = \frac{1}{2} |-3i + j|

  • a_{11} = \frac{1}{2} |-3(1) + 1| = \frac{1}{2} |-3 + 1| = \frac{1}{2} |-2| = \frac{1}{2}(2) = 1
  • a_{12} = \frac{1}{2} |-3(1) + 2| = \frac{1}{2} |-3 + 2| = \frac{1}{2} |-1| = \frac{1}{2}(1) = \frac{1}{2}
  • a_{13} = \frac{1}{2} |-3(1) + 3| = \frac{1}{2} |-3 + 3| = \frac{1}{2} |0| = 0
  • a_{14} = \frac{1}{2} |-3(1) + 4| = \frac{1}{2} |-3 + 4| = \frac{1}{2} |1| = \frac{1}{2}(1) = \frac{1}{2}
  • a_{21} = \frac{1}{2} |-3(2) + 1| = \frac{1}{2} |-6 + 1| = \frac{1}{2} |-5| = \frac{1}{2}(5) = \frac{5}{2}
  • a_{22} = \frac{1}{2} |-3(2) + 2| = \frac{1}{2} |-6 + 2| = \frac{1}{2} |-4| = \frac{1}{2}(4) = 2
  • a_{23} = \frac{1}{2} |-3(2) + 3| = \frac{1}{2} |-6 + 3| = \frac{1}{2} |-3| = \frac{1}{2}(3) = \frac{3}{2}
  • a_{24} = \frac{1}{2} |-3(2) + 4| = \frac{1}{2} |-6 + 4| = \frac{1}{2} |-2| = \frac{1}{2}(2) = 1
  • a_{31} = \frac{1}{2} |-3(3) + 1| = \frac{1}{2} |-9 + 1| = \frac{1}{2} |-8| = \frac{1}{2}(8) = 4
  • a_{32} = \frac{1}{2} |-3(3) + 2| = \frac{1}{2} |-9 + 2| = \frac{1}{2} |-7| = \frac{1}{2}(7) = \frac{7}{2}
  • a_{33} = \frac{1}{2} |-3(3) + 3| = \frac{1}{2} |-9 + 3| = \frac{1}{2} |-6| = \frac{1}{2}(6) = 3
  • a_{34} = \frac{1}{2} |-3(3) + 4| = \frac{1}{2} |-9 + 4| = \frac{1}{2} |-5| = \frac{1}{2}(5) = \frac{5}{2}

So, the matrix is:

A = \begin{bmatrix} 1 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{5}{2} & 2 & \frac{3}{2} & 1 \\ 4 & \frac{7}{2} & 3 & \frac{5}{2} \\ \end{bmatrix}

Part (ii): a_{ij} = 2i - j

  • a_{11} = 2(1) - 1 = 2 - 1 = 1
  • a_{12} = 2(1) - 2 = 2 - 2 = 0
  • a_{13} = 2(1) - 3 = 2 - 3 = -1
  • a_{14} = 2(1) - 4 = 2 - 4 = -2
  • a_{21} = 2(2) - 1 = 4 - 1 = 3
  • a_{22} = 2(2) - 2 = 4 - 2 = 2
  • a_{23} = 2(2) - 3 = 4 - 3 = 1
  • a_{24} = 2(2) - 4 = 4 - 4 = 0
  • a_{31} = 2(3) - 1 = 6 - 1 = 5
  • a_{32} = 2(3) - 2 = 6 - 2 = 4
  • a_{33} = 2(3) - 3 = 6 - 3 = 3
  • a_{34} = 2(3) - 4 = 6 - 4 = 2

So, the matrix is:

A = \begin{bmatrix} 1 & 0 & -1 & -2 \\ 3 & 2 & 1 & 0 \\ 5 & 4 & 3 & 2 \\ \end{bmatrix}

Common mistakes

  • Incorrectly identifying the order of a matrix (rows x columns).
  • Errors in calculating the total number of elements from the order.
  • Misinterpreting the indices (i, j) when identifying or constructing elements.
  • Forgetting to consider all factor pairs when determining possible matrix orders.

Revision tips

  • Review the definitions of matrix order and elements thoroughly.
  • Practice identifying elements using row and column indices (a_ij).
  • Work through examples of finding all possible orders for matrices with various numbers of elements.
  • Pay close attention to the formulas used for constructing matrices and substitute values carefully.

Practice MCQs

Q1. What is the order of a matrix with 12 elements?

Q2. If a matrix has 5 elements, what are its possible orders?

Q3. In a matrix A, if an element is denoted by a_23, it refers to the element in:

Q4. How many elements does a 3x4 matrix have?

Frequently asked questions

What is the main focus of the NCERT Solutions for Matrices (Chapter 3) for Class 12?

These solutions focus on understanding the fundamental concepts of matrices, including their order, the number of elements, how to identify specific elements, determining possible orders for a given number of elements, and constructing matrices based on given rules.

How do these solutions help in determining the order of a matrix?

The solutions explain that the order of a matrix is represented as 'rows × columns'. By counting the number of rows and columns in a given matrix, students can determine its order.

What does the notation a_ij represent in a matrix?

The notation a_ij represents the element located in the i-th row and the j-th column of the matrix.

How can I find the possible orders of a matrix if I know the number of elements?

To find the possible orders, you need to find all pairs of natural numbers (factors) whose product equals the total number of elements. Each pair (m, n) represents a possible order (m × n).

Are the solutions useful for exam preparation?

Yes, these solutions provide clear, step-by-step explanations and cover all types of problems typically asked in exams related to the basics of matrices, making them excellent for revision and practice.

Important links

Content reviewed by the NCERT Help team. Editorial Team and update policy

NCERT Solutions PDF PDF on NCERT Help. URL unchanged for search indexing.