NCERT Solutions Exercise 3.1: Matrices

# NCERT Solutions Class 12 Maths Chapter 3 - Matrices

### Exercise 3.1

Q1: In the matrix A= write:
(i)   The order of the matrix
(ii)  The number of elements,
(iii) Write the elements

Ans:
(i) In the given matrix, the number of rows is 3 and the number of columns is 4.
Therefore, the order of the matrix is 3 × 4.
(ii)  Since the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.
(iii)

Q2: If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
Ans: 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.

Q3: If a matrix has 18 elements, what are the possible order it can have? What, if it has 5 elements?
Ans: 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 two these are 1 × 5 and 5 × 1.

Q4: Construct a 2 × 2 matrix, A = [aij], whose elements are given by:

Ans: (i) Given that
In general a 2 × 2 matrix is given by

Therefore, the required matrix is
(ii) aij=
In general a 2 × 2 matrix is given by

Therefore, the required matrix is
(iii) Given that aij
In general a 2 × 2 matrix is given by

Therefore, the required matrix is

Q5: Construct a 3 × 4 matrix, whose elements are given by
(i) aij =
(ii) aij =
(i) aij =
In general a 3 × 4 matrix is given by aij =

(ii) aij

Q6: Find the value of x, y, and z from the following equation:
(i)

(ii)
(iii)
Ans:
(i)
As the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
x = 1, y = 4, and 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,
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 and y = 2
When x − y = − 2 and x + y = 6, we get x = 2 and y = 4
∴ x = 4, y = 2, and z = 0 or x = 2, y = 4, and z = 0
(iii)
As the two matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
x + y + z = 9 … (1)
x + z = 5 …….. (2)
y + z = 7 …….. (3)
From (1) and (2), we have:
y + 5 = 9
⇒y = 4
Then, from (3), we have:
4 + z = 7
⇒z = 3
∴ x + z = 5
⇒x = 2
∴ x = 2, y = 4 and z = 3.

Question 7: Find the value of a, b, c, and d from the equation:

Ans:
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.

Q8:is a square matrix, if
(A) m < n
(B) m > n
(C) m = n
(D) None of these
Ans: The correct answer is C.
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. Therefore,   is a square matrix, if m = n.

Q9: Which of the given values of x and y make the following pair of matrices equal

(A)
(B) Not possible to find
(C)
(D)
Ans: The correct answer is B.
It is given that
Equating the corresponding elements, we get:
3x+7=0 ⇒x = -7/3
and 5=y−2 ⇒y=7
y+1=8 ⇒ y = 7
and 2−3x=4 ⇒x = -2/3
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.

Q10: The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
(A)27
(B)18
(C)81
(D)512
Ans: 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.

The document NCERT Solutions Class 12 Maths Chapter 3 - Matrices is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on NCERT Solutions Class 12 Maths Chapter 3 - Matrices

 1. What are matrices and how are they used in mathematics?
Ans. Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are used to represent and work with data, perform operations like addition, subtraction, multiplication, and division, and solve systems of linear equations in mathematics.
 2. How do you add and subtract matrices?
Ans. To add or subtract matrices, you simply add or subtract the corresponding elements of the matrices. The matrices must have the same dimensions (same number of rows and columns) for addition or subtraction to be possible.
 3. What is the identity matrix and how is it used in matrix operations?
Ans. The identity matrix is a square matrix with 1s on the main diagonal (from the top left to the bottom right) and 0s elsewhere. When the identity matrix is multiplied with another matrix, it acts as the identity element for matrix multiplication and leaves the other matrix unchanged.
 4. How do you multiply matrices?
Ans. To multiply matrices A and B, the number of columns in matrix A must be equal to the number of rows in matrix B. The product matrix will have the same number of rows as matrix A and the same number of columns as matrix B. Each element in the product matrix is calculated by multiplying elements from the corresponding row of matrix A and the corresponding column of matrix B.
 5. Can matrices be used to solve systems of linear equations?
Ans. Yes, matrices can be used to solve systems of linear equations. By representing the coefficients of the variables and constants in a system of linear equations as a matrix, you can use matrix operations like Gaussian elimination or matrix inversion to find the solutions to the system.

## Mathematics (Maths) for JEE Main & Advanced

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