NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.1 Matrices

Class 12 Maths Chapter 3 Exercise 3.1 Matrices:- Matrices are a fundamental concept in mathematics, widely used in various fields such as engineering, physics, computer science, and economics. Chapter 3 of Class 12 Maths introduces students to matrices, their types, properties, and operations. A matrix is a rectangular array of numbers or functions arranged in rows and columns. It is a concise and systematic way to handle multiple equations and transformations simultaneously. The order of a matrix is defined by the number of rows (m) and columns (n) it has, represented as m×nm \times nm×n. 

For example, a matrix with 3 rows and 4 columns is of order 3×43 \times 43×4. Different types of matrices include row matrix, column matrix, square matrix, diagonal matrix, scalar matrix, identity matrix, and zero matrix. Basic concepts such as the equality of matrices (where two matrices are equal if they have the same order and their corresponding elements are identical) and the notation used (elements of a matrix AAA denoted as aija_{ij}aij​, where iii represents the row number and jjj represents the column number) are also introduced. Check out the 

Class 12 Maths Chapter 3 Exercise 3.1 Matrices below.

NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.1 Matrices

Question1. In the matrix A =, write:

(i) The order of the matrix.

(ii) The number of elements.

(iii) Write the elements

Solution :
(i) There are 3 horizontal lines (rows) and 4 vertical lines (columns) in the given matrix A.

Therefore, Order of the matrix is 3 x 4.

(ii) The number of elements in the matrix A is 3 x 4 = 12.

(iii) a13 Element in first row and third column = 19

a21 Element in second row and first column = 35

a33 Element in third row and third column = -5

a24 Element in second row and fourth column = 12

a23 Element in second row and third column = 5/2

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Question2. If a matrix has 24 elements, what are possible orders it can order? What, if it has 13 elements?

Solution :
We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 24 elements, we have to find all the ordered pairs of natural numbers whose product is 24.

The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and (6, 4)

Hence, the possible orders of a matrix having 24 elements are:

1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, and 6 × 4

(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.

Hence, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.

Question3. If a matrix has 18 elements, what are the possible orders it can have? What if has 5 elements?

Solution :
We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 18 elements, we have to find all the ordered pairs of natural numbers whose product is 18.

The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6,), and (6, 3)

Hence, the possible orders of a matrix having 18 elements are:

1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, and 6 × 3

(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.

Hence, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.

Question4. Construct a 2 x 2 matrix A = [aij ] whose elements are given by:

(i)

(ii) aij = i/j

(iii)

Solution :
(i)

(ii)

(iii)

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Question 5. Construct a 3 x 4 matrix, whose elements are given by:

(i)

(ii) aij = 2i – j

Solution :
(i) 

(ii)

Question6. Find the values of x,y and z from the following equations:

(i)

(ii)

(iii)

Solution :
(i)Given: 

As the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

x = 1, y = 4 = z = 3

(ii)

As the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

x + y = 6, xy = 8 and 5 + z = 5

Now, 5 + z = 5 ⇒ z = 0

We know that:

(x – y)2 = (x + y)2 – 4xy

⇒  (x – y)2 = 36 – 32 = 4

⇒x – y = ± 2

Now, when x – y = 2 and x + y = 6  , we get  x = 4, y = 2

When x – y = -2 and x + y = 6, we get  x = 2, y = 4

∴ x  = 4, y = 2, and z = 0,  or x = 2, y = 4 and z = 0

(iii)

Question7. Find the values of a,b, c and d from the equation .

Solution :

As the two matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

a − b = −1 … (1)

2a − b = 0 … (2)

2a + c = 5 … (3)

3c + d = 13 … (4)

From (2), we have:

b = 2a

Then, from (1), we have:

a − 2a = −1

⇒ a = 1

⇒ b = 2

Now, from (3), we have:

2 ×1 + c = 5

⇒ c = 3

From (4) we have:

3 ×3 + d = 13

⇒ 9 + d = 13 ⇒ d = 4

∴a = 1, b = 2, c = 3, and d = 4

Question 8. A = [aij]m x n is a square matrix if:

(A) m < n (B) m > n  (C) m = n (D) None of these

Solution :
It is known that a given matrix is said to be a square matrix if the number of rows is equal to the number of columns. A = [aij]m x n

option (C) is correct.

Question9. Which of the given values of  x and y make the following pairs of matrices equal:

(B) Not possible to find 

Solution :

We find that on comparing the corresponding elements of the two matrices, we get two different values of x , which is not possible.

Hence, it is not possible to find the values of  x and y  for which the given matrices are equal.

Therefore, option (B) is correct.

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Question10. The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is:

(A) 27

(B) 18

(C) 81

(D) 512

Solution :
The correct answer is D.

The given matrix of the order 3 × 3 has 9 elements and each of these elements can be either 0 or 1.

Now, each of the 9 elements can be filled in two possible ways.

Therefore, by the multiplication principle, the required number of possible matrices is 29 = 512

Class 12 Maths Chapter 3 Exercise 3.1 Matrices Summary

Definition of a Matrix:

• A matrix is a rectangular array of numbers arranged in rows and columns.

• It is usually denoted by a capital letter such as A, B, CA, B, CA,B, C, etc.

• Elements of a matrix are represented by aija_{ij}aij​ where iii is the row number and jjj is the column number.

Types of Matrices:

• Row Matrix: A matrix with a single row.

• Column Matrix: A matrix with a single column.

• Square Matrix: A matrix with the same number of rows and columns.

• Diagonal Matrix: A square matrix where all elements except the diagonal elements are zero.

• Scalar Matrix: A diagonal matrix where all the diagonal elements are equal.

• Identity Matrix: A square matrix with 1s on the diagonal and 0s elsewhere.

• Zero Matrix: A matrix where all elements are zero.

Order of a Matrix:

• The order of a matrix is defined by the number of rows and columns it has represented as m×nm \times nm×n.

Equality of Matrices:

• Two matrices are equal if they have the same order and corresponding elements are equal.

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Class 12 Maths Chapter 3 Exercise 3.1 Matrices FAQs

Q1. What is a matrix?

Ans. A matrix is a rectangular array of numbers arranged in rows and columns. It is denoted by a capital letter and the elements are referred to as aija_{ij}aij​ where iii is the row number and jjj is the column number.

Q2. What is the order of a matrix?

Ans. The order of a matrix is given by the number of rows and columns it contains, expressed as m×nm \times nm×n where mmm is the number of rows and nnn is the number of columns.

Q3. How do you determine if two matrices are equal?

Ans. Two matrices are equal if they have the same order and their corresponding elements are identical.

Q4. What is a row matrix?

Ans. A row matrix is a matrix that has only one row.

Q5. What is a column matrix?

Ans. A column matrix is a matrix that has only one column.