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

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

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

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

Construct a 2 × 2 matrix A = [a_{ij}] whose elements a_{ij}

are given by:

A_{ij}

= e^{2ix} sin xj

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

(i) aij = i + j

(ii) aij = i – j

(iii) aij = 2i

(iv) aij = j

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.

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

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

(ii)

Find x, y satisfying the matrix equations

(i)

(ii)

(i)

(ii)

If X and Y are 2 × 2 matrices, then

solve the following matrix equations for X and Y.

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

Use this to find A^{4}

Find the matrix A such that

Find the matrix A such that

So,

A^{16} is null matrix.

Then show that (A + B)^{2} = A^{2}

+ B^{2}.

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

3.

Is (AB)^{2} = A^{2} B^{2}?

Give reasons.

If A and B be square matrices of the same order such that AB = BA,

then show that (A + B)^{2} = A^{2} +

2AB + B^{2}.

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.

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?

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?

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

If l_{i}, m_{i}, n_{i} ; i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AA^{T} = I,

## Chapter 5 – Algebra of Matrices Exercise Ex. 5.5

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

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

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 .

If *A* is a square matrix such that *A*^{2}=*A*, then

write the value of 7 *A*-(*I*+*A*)^{3}, where *I*

is the identity matrix.

A^{2} = A

A^{3} = A^{2}

= A

7A – (I + A)^{3}

= 7A – (I^{3}

+ A^{3} + 3A^{2}I + 3AI^{2})

= 7A – (I + A + 3A +

3A)

= 7A – (I + 7A)

= –I

Write 2×2 matrix which is both

symmetric and skew-symmetric.

Construct a 2 × 2 matrix A = [aij] whose

elements aij are given by

## Chapter 5 – Algebra of Matrices Exercise MCQ

a. a null matrix

b. a unit matrix

c. -A

d. A

a.

b.

c.

d.

If A and B are two matrices such that AB = A and BA = B, then B^{2} is equal to

a. B

b. A

c. 1

d. 0

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

a. B^{2} = B and A^{2} = A

b. B^{2}≠ B and A^{2} = A

c. A^{2}≠ A,B^{2} = B

d. A^{2}≠ A, B^{2}≠ B

If A and B are two matrices such that AB = B and BA =A, then A^{2} + B^{2} is equal to

a. 2 AB

b. 2 BA

c. A + B

d. AB

a. 3

b. 4

c. 6

d. 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

a.

b.

c.

d.

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

a. B

b. nB

c. B^{n}

d. A + B

a.

b.

c.

d.

a. 0

b. -1

c. 2

d. None of these

a. a = 4, b = 1

b. a = 1, b = 4

c. a = 0, b = 4

d. a = 2, b = 4

a. 1 + α^{2} + βγ = 0

b. 1 – α^{2} + βγ = 0

c. 1 – α^{2} – βγ = 0

d. 1 + α^{2} – βγ = 0

If S = [s_{ij}] is a scalar matrix such that s_{ii} = k and A is a square matrix of the same order, then AS = SA = ?

a. A^{k}

b. k + A

c. kA

d. kS

If A is a square matrix such that A^{2} = A, then (I + A)^{3} – 7A is equal to

a. A

b. I – A

c. I

d. 3A

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

a. A skew-symmetric matrix

b. A symmetric matrix

c. A diagonal matrix

d. An upper triangular matrix

If A is a square matrix, then AA is a

a. Skew-symmetric matrix

b. Symmetric matrix

c. Diagonal matrix

d. None of these

If A and B are symmetric matrices, then ABA is

a. Symmetric matrix

b. Skew-symmetric matrix

c. Diagonal matrix

d. Scalar matrix

a. X = 0, y = 0

b. X + y = 5

c. X = y

d. None of these

If A is 3 × 4 matrix and B is a matrix such that A^{T}B and BA^{T} are both defined. Then, B is of the type

a. 3 × 4

b. 3 × 3

c. 4 × 4

d. 4 × 3

If A = [a_{ij}] is a square matrix of even order such that a_{ij} = i^{2} – j^{2}, 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

a.

b.

c.

d. none of these

a.

b.

c.

d.

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

a.

b.

c.

d.

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

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

a.

b.

c.

d. Not possible to find

a. -6, -12, -18

b. -6, 4, 9

c. -6, -4, -9

d. -6, 12, 18

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

a. 17

b. 25

c. 3

d. 12

If A = [a_{ij}] is a scalar matrix of order n × n such that a_{ii} = k for all I, then trace of A is equal to

a. nk

b. n + k

c.

d. none of these

a. square matrix

b. diagonal matrix

c. unit matrix

d. none of these

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

a. x = 3, y = 1

b. x = 2, y = 3

c. x = 2, y = 3

d. x = 3, y = 3

If A is a square matrix such that A^{2} = I, then (A – I)^{3} + (A + I)^{3} – 7A is equal to

a. A

b. I – A

c. I + A

d. 3A

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

If A is a matrix of order m × n and B is a matrix such that AB^{T} and B^{T}A are both defined, then the order of matrix B is

a. m × n

b. n × n

c. n × m

d. 3 × n

If A and B are matrices of the same order, then (AB^{T}-BA^{T})^{T} is a

a. skew-symmetric matrix

b. null matrix

c. unit matrix

d. symmetric matrix

a. I

b. A

c. O

d. –I * *

a. I

b. 0

c. 2I

d.

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

a. A^{2} – B^{2}

b. A^{2} – BA – AB – B^{2}

c. A^{2} – B^{2} + BA – AB

d. A^{2} – BA + B^{2} + AB

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

a. Diagonal matrix

b. Symmetric matrix

c. Skew-symmetric matrix

d. Scalar matrix

a. Identity matrix

b. Symmetric matrix

c. Skew-symmetric matrix

d. Diagonal matrix

Correct option: (d)

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