NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

Mathematics (Maths) Class 12

Class 12 : NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

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Exercise 3.1

Question 1: In the matrix A=NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev write:
 (i)   The order of the matrix
 (ii)  The number of elements,
 (iii) Write the elements
  NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
Answer:
(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)  NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

Question 2: If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
Answer:
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.

Question 3: If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Answer:
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.

Question 4: Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
1.NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

2.NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

3. NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
Answer:
(i) Given that  NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

In general a 2 × 2 matrix is given by  NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

Therefore, the required matrix is   NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
(ii) aij= NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

In general a 2 × 2 matrix is given by   NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

Therefore, the required matrix is  NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev


(iii) Given that aij  NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

In general a 2 × 2 matrix is given by   NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

Therefore, the required matrix is  NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

Question 5: Construct a 3 × 4 matrix, whose elements are given by
(i) aij =NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

(ii) aij =NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
Answer:
(i) aij = NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
In general a 3 × 4 matrix is given by aij = NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev


(ii) aij
NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

Question 6: Find the value of x, y, and z from the following equation:
 (i)
NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

(ii) NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

(iii) NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
Answer:
(i) NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

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) NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
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) NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
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:
NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

Answer:
NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
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:NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRevis a square matrix, if
(A)m < n
(B)m > n
(C)m = n
(D)None of these
Answer:
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, NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev  is a square matrix, if m = n.

Question 9: Which of the given values of x and y make the following pair of matrices equal 

NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev   NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev
Answer:
The correct answer is B.

It is given that  NCERT Solutions - Matrices, Exercise 3.1 JEE Notes | EduRev

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.

Question 10: 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
Answer:
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.

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