## EXERCISE 3.1

#### Page No 64:

#### Question 1:

In the matrix, write:

(i) The order of the matrix (ii) The number of elements,

(iii) Write the elements *a*_{13}, *a*_{21}, *a*_{33}, *a*_{24}, *a*_{23}

#### 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)** *a*_{13} = 19, *a*_{21} = 35, *a*_{33} = −5, *a*_{24} = 12, *a*_{23} =

#### 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,, whose elements are given by:

(i)

(ii)

(iii)

#### Answer:

In general, a 2 × 2 matrix is given by

**(i)**

Therefore, the required matrix is

**(ii)**

Therefore, the required matrix is

**(iii)**

Therefore, the required matrix is

#### Question 5:

Construct a 3 × 4 matrix, whose elements are given by

(i) (ii)

#### Answer:

In general, a 3 × 4 matrix is given by

**(i)**

Therefore, the required matrix is

**(ii)**

Therefore, the required matrix is

#### Question 6:

Find the value of *x*, *y*, and *z* from the following equation:

(i) (ii)

(iii)

#### Answer:

**(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} − 4*xy*

⇒ (*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:

#### Answer:

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

Comparing the corresponding elements, we get:

*a* − *b* = −1 … (1)

2*a* − *b* = 0 … (2)

2*a* + *c* = 5 … (3)

3*c* + *d* = 13 … (4)

From (2), we have:

*b* = 2*a*

Then, from (1), we have:

*a *− 2*a* = −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

#### Page No 65:

#### Question 8:

is 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, 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

**(A)**

**(B)** Not possible to find

**(C)**

**(D)**

#### Answer:

The correct answer is B.

It is given that

Equating the corresponding elements, we get:

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 2^{9} = 512