# RD Sharma Solution CLass 12 Mathematics Chapter 5 Algebra of Matrices

## Chapter 5 – Algebra of Matrices Exercise Ex. 5.1

Question 1

If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

Solution 1

We know that if a matrix is of the order , it has mn elements. Thus, to find all the possible orders of a matrix having 8 elements, we have to find all the ordered pairs of natural numbers whose products is 8.

The ordered pairs are:

are the ordered pairs of natural numbers whose product is 5.

Hence, the possible orders of a matrix having 5 elements are

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

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

Solution 5

Question 6

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

Aij
= e2ix sin xj

Solution 6

Question 7

Construct a 3 × 4 matrix A = [aij] whose elements aij are given by :

(i) aij = i + j

(ii) aij = i – j

(iii) aij = 2i

(iv) aij = j

Solution 7

(v)
Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

The sales figure of two car dealer during january 2013 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars, while dealer B sold 7 deluxe, 2 premium and 3 standard cars. Total sales over the 2 month period of january – february revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars. in the same 2 month period, dealer b sold 10 deluxe, 5 premium and 7 standard cars. Write 2 × 3 matrices summarizing for january and 2 – month period for each dealer.

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Find the values of a and b if A = B, where

Solution 22

## Chapter 5 – Algebra of Matrices Exercise Ex. 5.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

(ii)

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Find x, y satisfying the matrix equations

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

(i)

(ii)

Solution 23

(i)

(ii)

Question 24

Solution 24

Question 25

Solution 25

Question 26

If X and Y are 2 × 2 matrices, then
solve the following matrix equations for X and Y.

Solution 26

Question 27

Solution 27

## Chapter 5 – Algebra of Matrices Exercise Ex. 5.3

Question 1

Solution 1
Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Use this to find A4

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Question 63

Find the matrix A such that

Solution 63

Question 64

Find the matrix A such that

Solution 64

Question 65

Solution 65

Question 66

Solution 66

So,

A16 is null matrix.

Question 67

Then show that (A + B)2 = A2
+ B2.

Solution 67

Question 68

Solution 68

Question 69

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

Solution 72

Question 73

Solution 73

Question 74

Solution 74

Question 75

Solution 75

Question 76

Solution 76

Question 77

Solution 77

Question 78

Solution 78

Question 79

Solution 79

Question 80

Solution 80

Question 81

Solution 81

Question 82

Solution 82

Question 83

Solution 83

Question 84

Solution 84

Question 85

Solution 85

Question 86

Solution 86

Question 87

Let A and B be square matrices of the order 3 ×
3.

Is (AB)2 = A2 B2?
Give reasons.

Solution 87

Question 88

If A and B be square matrices of the same order such that AB = BA,
then show that (A + B)2 = A2 +
2AB + B2.

Solution 88

Question 89

Solution 89

Question 90

Solution 90

Question 91

Solution 91

Question 92

Solution 92

Question 93

Solution 93

Question 94

To promote making of toilets for women, an organization tried to
generate awareness through (i) house calls (ii)
letters and (iii) announcements. The cost for each mode per attempt is given
below:

i. Rs. 50

ii. Rs. 20

iii. Rs. 40

The
number of attempts made in three villages X, Y, and Z are given below:

 (i) (ii) (iii) X 400 300 100 Y 300 250 75 z 500 400 150

Find
the total cost incurred by the organization for three villages separately,
using matrices.

Solution 94

Question 95

There are 2 families A and B. There are 4 men, 6 women and 2 children
in family A, and 2 men, 2 women and 4 children in family B. The recommend
daily amount of calories is 2400 for men, 1900 for women, 1800 for children
and 45 grams of proteins for men, 55 grams for women and 33 grams for
children. Represent the above information using matrix. Using matrix
multiplication, Calculate the total requirement of calories and proteins for
each of the families. What awareness can you create among people about the
planned diet from this question?

Solution 95

Question 96

In a parliament election, a political party hired a public relations
firm to promote its candidates in three ways – telephone, house calls and
letters. The cost per contact (in paisa) is given in matrix A as

The number of contacts of each type made in two cities X and Y is
given in the matrix B as

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote – party’s promotional activity or
their social activities?

Solution 96

## Chapter 5 – Algebra of Matrices Exercise Ex. 5.4

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

If li, mi, ni ; i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AAT = I,

Solution 16

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4

Question 5
Solution 5

Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8

## Chapter 5 – Algebra of Matrices Exercise Ex. 5VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

If a matrix has 5 elements, write all possible orders it can have.

Solution 49

If a matrix is of order , then the number of elements in the matrix is the product .

Given that the required matrix is having 5 elements and 5 is a prime number.

Hence the prime factorization of 5 is either .

Thus, the order of the matrix is either .

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

If A is a square matrix such that A2=A, then
write the value of 7 A-(I+A)3, where I
is the identity matrix.

Solution 56

A2 = A

A3 = A2
= A

7A – (I + A)3

= 7A – (I3
+ A3 + 3A2I + 3AI2)

= 7A – (I + A + 3A +
3A)

= 7A – (I + 7A)

= I

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Write 2×2 matrix which is both
symmetric and skew-symmetric.

Solution 60

Question 61

Solution 61

Question 62

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

Solution 62

Question 63

Solution 63

## Chapter 5 – Algebra of Matrices Exercise MCQ

Question 1

a. a null matrix

b. a unit matrix

c. -A

d. A

Solution 1

Question 2

a.

b.

c.

d.

Solution 2

Question 3

If A and B are two matrices such that AB = A and BA = B, then B2 is equal to

a. B

b. A

c. 1

d. 0

Solution 3

Question 4

If AB = A and BA = B, where A and B are square matrices, then

a. B2 = B and A2 = A

b. B2≠ B and A2 = A

c. A2≠ A,B2 = B

d. A2≠ A, B2≠ B

Solution 4

Question 5

If A and B are two matrices such that AB = B and BA =A, then A2 + B2 is equal to

a. 2 AB

b. 2 BA

c. A + B

d. AB

Solution 5

Question 6

a. 3

b. 4

c. 6

d. 7

Solution 6

Question 7

If the matrix AB is zero , then

a. It is not necessary that either A = 0 or B = 0

b. A = 0 or B = 0

c. A = O and B = 0

d. All the above statements are wrong

Solution 7

Question 8

a.

b.

c.

d.

Solution 8

Question 9

If A, B are square matrices of order 3, A is non-singular and AB = 0, then B is a

a. Null matrix

b. Singular matrix

c. Unit matrix

d. Non-singular matrix

Solution 9

Question 10

a. B

b. nB

c. Bn

d. A + B

Solution 10

Question 11

a.

b.

c.

d.

Solution 11

Question 12

a. 0

b. -1

c. 2

d. None of these

Solution 12

Question 13

a. a = 4, b = 1

b. a = 1, b = 4

c. a = 0, b = 4

d. a = 2, b = 4

Solution 13

Question 14

a. 1 + α2 + βγ = 0

b. 1 – α2 + βγ = 0

c. 1 – α2 – βγ = 0

d. 1 + α2 – βγ = 0

Solution 14

Question 15

If S = [sij] is a scalar matrix such that sii = k and A is a square matrix of the same order, then AS = SA = ?

a. Ak

b. k + A

c. kA

d. kS

Solution 15

Question 16

If A is a square matrix such that A2 = A, then (I + A)3 – 7A is equal to

a. A

b. I – A

c. I

d. 3A

Solution 16

Question 17

If a matrix A is both symmetric and skew-symmetric, then

a. A is a diagonal matrix

b. A is a zero matrix

c. A is a scalar matrix

d. A is a square matrix

Solution 17

Question 18

a. A skew-symmetric matrix

b. A symmetric matrix

c. A diagonal matrix

d. An upper triangular matrix

Solution 18

Question 19

If A is a square matrix, then AA is a

a. Skew-symmetric matrix

b. Symmetric matrix

c. Diagonal matrix

d. None of these

Solution 19

Question 20

If A and B are symmetric matrices, then ABA is

a. Symmetric matrix

b. Skew-symmetric matrix

c. Diagonal matrix

d. Scalar matrix

Solution 20

Question 21

a. X = 0, y = 0

b. X + y = 5

c. X = y

d. None of these

Solution 21

Question 22

If A is 3 × 4 matrix and B is a matrix such that ATB and BAT are both defined. Then, B is of the type

a. 3 × 4

b. 3 × 3

c. 4 × 4

d. 4 × 3

Solution 22

Question 23

If A = [aij] is a square matrix of even order such that aij = i2 – j2, then

a. A is a skew – symmetric matrix and |A| = 0

b. A is symmetric matrix and |A| is a square

c. A is symmetric matrix and |A| = 0

d. None of these

Solution 23

Question 24

a.

b.

c.

d. none of these

Solution 24

Question 25

a.

b.

c.

d.

Solution 25

Question 26

Out of the following matrices, choose that matrix which is a scalar matrix:

a.

b.

c.

d.

Solution 26

Question 27

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

Solution 27

Question 28

Which of the given values of x and y make the following pairs of matrices equal?

a.

b.

c.

d. Not possible to find

Solution 28

Question 29

a. -6, -12, -18

b. -6, 4, 9

c. -6, -4, -9

d. -6, 12, 18

Solution 29

Question 30

a. I cos  θ + J sin θ

b. I sin θ + J cos θ

c. I cos θ – J sin θ

d. – I cos θ + J sin θ

Solution 30

Question 31

a. 17

b. 25

c. 3

d. 12

Solution 31

Question 32

If A = [aij] is a scalar matrix of order n × n such that aii = k for all I, then trace of A is equal to

a. nk

b. n + k

c.

d. none of these

Solution 32

Question 33

a. square matrix

b. diagonal matrix

c. unit matrix

d. none of these

Solution 33

Question 34

The number of possible matrices of order 3 × 3 with each entry 2 or 0 is

a. 9

b. 27

c. 81

d. none of these

Solution 34

Question 35

a. x = 3, y = 1

b. x = 2, y = 3

c. x = 2, y = 3

d. x = 3, y = 3

Solution 35

Question 36

If A is a square matrix such that A2 = I, then (A – I)3 + (A + I)3 – 7A is equal to

a. A

b. I – A

c. I + A

d. 3A

Solution 36

Question 37

If A and B are two matrix of order 3 × m and 3 × n respectively and m = n, then the order of 5A – 2B is

a. m × n

b. 3 × 3

c. m × n

d. 3 × n

Solution 37

Question 38

If A is a matrix of order m × n and B is a matrix such that ABT and BTA are both defined, then the order of matrix B is

a. m × n

b. n × n

c. n × m

d. 3 × n

Solution 38

Question 39

If A and B are matrices of the same order, then (ABT-BAT)T is a

a. skew-symmetric matrix

b. null matrix

c. unit matrix

d. symmetric matrix

Solution 39

Question 40

a. I

b. A

c. O

d. I

Solution 40

Question 41

a. I

b. 0

c. 2I

d.

Solution 41

Question 42

If A and B are square matrices of the same order, then (A + B) (A – B) is equal to

a. A2 – B2

b. A2 – BA – AB – B2

c. A2 – B2 + BA – AB

d. A2 – BA + B2 + AB

Solution 42

Question 43

a. Only AB is defined

b. Only BA is defined

c. AB ad BA both are defined

d. AB and BA both are not defined

Solution 43

Question 44

a. Diagonal matrix

b. Symmetric matrix

c. Skew-symmetric matrix

d. Scalar matrix

Solution 44

Question 45

a. Identity matrix

b. Symmetric matrix

c. Skew-symmetric matrix

d. Diagonal matrix

Solution 45

Correct option: (d)

A matrix is called Diagonal matrix if all the elements, except those in the leading diagonal, are zero.

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