NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

JEE: NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

The document NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
All you need of JEE at this link: JEE

Exercise 3.1 

Question 1: In the matrix A=NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE write:
 (i)   The order of the matrix
 (ii)  The number of elements,
 (iii) Write the elements
  NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
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 Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

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 Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

2.NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

3. NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
Answer:
(i) Given that  NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

In general a 2 × 2 matrix is given by  NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

Therefore, the required matrix is   NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
(ii) aij= NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

In general a 2 × 2 matrix is given by   NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

Therefore, the required matrix is  NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE


(iii) Given that aij  NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

In general a 2 × 2 matrix is given by   NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

Therefore, the required matrix is  NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

Question 5: Construct a 3 × 4 matrix, whose elements are given by
(i) aij =NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

(ii) aij =NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
Answer:
(i) aij = NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
In general a 3 × 4 matrix is given by aij = NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE


(ii) aij
NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

Question 6: Find the value of x, y, and z from the following equation:
 (i)
NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

(ii) NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

(iii) NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
Answer:
(i) NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

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 Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
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 Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
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 Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

Answer:
NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
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 Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEEis 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 Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE  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 Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
(A)NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
(B) Not possible to find
(C) NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
(D) NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE
Answer:
The correct answer is B.

It is given that  NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE

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.

The document NCERT Solutions Exercise 3.1: Matrices Notes | Study Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
All you need of JEE at this link: JEE

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