##### Question 1:

In the matrix

(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) The order of the matrix = 3 x 4.

(ii) The number elements = 3 x 4 = 12.

##### Question 2:

If a matrix has 24 elements, what are the possible orders it can have ? What, if it has 13 elements ?

##### Answer:

(i) Since

24 = 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6,

$\therefore $ there are 8 matrices having 24 elements.

Their orders are 1 x 24, 24 x 1, 2 x 12, 12 x 2, 3 x 8, 8 x 3, 4 x 6, 6 x 4.

(ii) Since 13 = 1 x 13,

$\therefore $ there are 2 matrices having 13 elements.

Their orders are 1 x 13 and 13 x 1.

##### Question 3:

If a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements ?

##### Answer:

(i) Since 18 = 1 x 18 = 2 x 9 = 3 x 6,

$\therefore $ there are 6 matrices having 18 elements.

Their oders are 1 x 18, 18 x 1, 2 x 9, 9 x 2, 3 x 6, 6 x 3.

(ii) Since 5 = 1 x 5,

$\therefore $ there are 2 matrices having 5 elements.

Their orders are 1 x 5 and 5 x 1.

##### Question 4:

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

##### Answer:

##### Question 5:

Construct a 3 x 4 matrix whose elements are given by :

##### Answer:

##### Question 6:

Find the values of x, y and z from the following equations :

##### Answer:

Comparing corresponding elements :

x + y = 6 ...(1)

5 + z = 5 ...(2)

and xy = 8 ...(3)

From (1), y = 6 - x ...(4)

Putting in (3), x (6 - x) = 8

$\Rightarrow $ x^{2} - 6x + 8 = 0

$\Rightarrow $ (x - 2) (x - 4) = 0

$\Rightarrow $ x = 2, 4.

From (4), y = 6 - 2 = 4, 6 - 4 = 2.

Also from (2), z =0.

Hence, x = 2, y = 4, z = 0

or x = 4, y = 2, z = 0.

Comparing corresponding elements :

x + y + z = 9 ...(1)

x + z = 5 ...(2)

and y + z = 7 ...(3)

Subtracting (3) from (1),

x =2.

Subtracting (2) from (1),

y = 4.

Putting in (1), 2 + 4 + z = 9

$\Rightarrow $ z = 9 - 6 = 3.

Hence, x = 2, y = 4, and z = 3.

##### Question 7:

Find the values of a, b, c and d from the equation :

##### Answer:

We have :

Comparing corresponding elements :

a - b = - 1 ...(1)

2a + c = 5 ...(2)

2a - b = 0 ...(3)

and 3c + d = 13 ...(4)

Subtracting (1) from (3),

a =1.

Putting in (3), 2(1) - b =0

$\Rightarrow $ b =2.

Putting in (2),2 (1) + c =5

$\Rightarrow $ c = 5 - 2 = 3.

Putting in (4),3 (3) + d =13

$\Rightarrow $ d = 13 - 9 = 4.

Hence, a = 1, b = 2, c = 3 and d = 4.

##### Question 8:

A = [a_{ij}]_{m x n} is a square matrix, if :

- m < n
- m > n
- m = n
- None of these

##### Answer:

m = n

##### Question 9:

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

- Not possible to find

##### Answer:

Not possible to find

Reason : Comparing coresponding elements :

3x + 7 = 0 ...(1)

y - 2 = 5 ...(2)

y + 1 = 8 ...(3)

and 2 - 3x = 4 ...(4)

Thus x has two values.

Hence, it is not possible to find the values of x and y.

##### Question 10:

The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is :

- 27
- 18
- 81
- 512

##### Answer:

512

##### Question 11:

A + B

##### Answer:

##### Question 12:

A - B

##### Answer:

##### Question 13:

3A - C

##### Answer:

##### Question 14:

AB

##### Answer:

##### Question 15:

BA.

##### Answer:

##### Question 16:

Compute the following :

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##### Question 17:

Compute the following :

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##### Question 18:

Compute the following :

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##### Question 19:

Compute the following :

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##### Question 20:

Compute the indicated products :

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##### Question 21:

Compute the indicated products :

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##### Question 22:

Compute the indicated products :

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##### Question 23:

Compute the indicated products :

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##### Question 24:

Compute the indicated products :

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##### Question 25:

Compute the indicated products :

##### Answer:

##### Question 26:

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##### Question 27:

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##### Question 28:

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##### Question 29:

Find X and Y, if :

##### Answer:

##### Question 30:

Find X and Y, if :

##### Answer:

##### Question 31:

##### Answer:

##### Question 32:

##### Answer:

##### Question 33:

Solve the equation for x, y, z and t, if :

##### Answer:

We have :

Equating corresponding elements :

2x + 3 = 9 $\Rightarrow $ 2x = 9 - 3 = 6 $\Rightarrow $ x = 3

2y = 12 $\Rightarrow $ y = 6

2z - 3 = 15 $\Rightarrow $ 2z = 3 + 15 = 18 $\Rightarrow $ z = 9

and 2t + 6 = 18 $\Rightarrow $ 2t = 18 - 6 = 12 $\Rightarrow $ t = 6.

Hence, x = 3, y = 6, z = 9 and t = 6.

##### Question 34:

##### Answer:

We have :

Equating corresponding elements :

2x - y = 10 ....(1)

and 3x + y = 5 ....(2)

Adding (1) and (2), 5x = 15 $\Rightarrow $ x = 3.

Putting in (1), 2 (3) - y = 10 $\Rightarrow $ y = 6 - 10 = - 4.

Hence, x = 3 and y = - 4.

##### Question 35:

##### Answer:

Equating corresponding elements :

3x = x + 4 $\Rightarrow $ 2x = 4 $\Rightarrow $ x = 2

3y = x + y + 6 $\Rightarrow $ 2y = 2 + 6 $\Rightarrow $ 2y = 8 $\Rightarrow $ y = 4

3w =2w + 3 $\Rightarrow $ w = 3

and 3z = z + w - 1 $\Rightarrow $ 2z = 3 - 1 = 2 $\Rightarrow $ z = 1.

Hence, x = 2, y = 4, z = 1 and w = 3.

##### Question 36:

Show that f (x) f (y) = f (x + y).

##### Answer:

##### Question 37:

Show that :

##### Answer:

##### Question 38:

Show that :

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##### Question 39:

##### Answer:

##### Question 40:

##### Answer:

##### Question 41:

##### Answer:

##### Question 42:

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##### Question 43:

A trust fund has ₹30,000 that must be invested in two different types of bonds. The first bond pays 5% and second 7% interest per year. Using matrix multiplication, determine how to divide ₹30,000 among the two types of bonds if the trust fund must obtain an annual total interest of :

(a) ₹1800 (b) ₹2000.

##### Answer:

Let ₹30,000 be divided into two parts :

₹ x invested in 1st type and ₹(30,000 - x) in 2nd type.

The values of the bonds are represented by 1 x 2 row matrix as :

A = [ x 30,000 − x ] .

The amount received as interest per ₹ annually are represented by 2 x 1 column matrix as :

##### Question 44:

The book shop of a particular school has 10 dozen Chemistry books, 8 dozen Physics books, 10 dozen Economics books. The selling prices are ₹80, ₹60 and ₹40 each respectively. Find the total amount, the book-shop will receive from selling all the books, using matrix algebra.

##### Answer:

Inventory,

= [120 x 80 + 96 x 60 + 120 x 40]= [9600 + 5760 + 4800] = [20160].

Hence, the book-shop will receive ₹ 20,160 by selling all the books.

Assume X, Y, Z, W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3 and p x k respectively. Choose the correct answer in (21 - 22) :

##### Question 45:

The restriction on n, k and p so that PY + WY will be defined as :

- k = 3, p = n
- k is arbitrary, p = 2
- p is arbitrary, k = 3
- k = 2, p = 3.

##### Answer:

k = 3, p = n

##### Question 46:

If n = p, then the order of 7X - 5Z is :

- p x 2
- 2 x n
- n x 3
- p x n.

##### Answer:

2 x n

##### Question 47:

Find the transpose of each of the following matrices :

##### Answer:

##### Question 48:

Find the transpose of each of the following matrices :

##### Answer:

##### Question 49:

Find the transpose of each of the following matrices :

##### Answer:

##### Question 50:

##### Answer:

##### Question 51:

##### Answer:

##### Question 52:

##### Answer:

We have :

##### Question 53:

For the matrices A and B, verify that :

(AB)′ = B′A′, where :

##### Answer:

##### Question 54:

For the matrices A and B, verify that :

(AB)′ = B′A′, where :

##### Answer:

##### Question 55:

##### Answer:

##### Question 56:

##### Answer:

##### Question 57:

Show that the matrix

##### Answer:

We have :

##### Question 58:

Show that the matrix

##### Answer:

We have :

##### Question 59:

(i) A + A′ is a symmetric matrix

(ii) A - A′ is a skew-symmetric matrix.

##### Answer:

We have :

##### Question 60:

##### Answer:

##### Question 61:

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix :

##### Answer:

##### Question 62:

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix :

##### Answer:

##### Question 63:

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix :

##### Answer:

##### Question 64:

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix :

##### Answer:

##### Question 65:

Choose the correct answer

If A, B are symmetric matrices of same order, then AB - BA is a :

- Skew-symmetric matrix
- Symmetric matrix
- Zero matrix
- Identity matrix

##### Answer:

Skew-symmetric matrix

##### Question 66:

Choose the correct answer

- π

##### Answer:

##### Question 67:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 68:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 69:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 70:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 71:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 72:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 73:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 74:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 75:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 76:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 77:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 78:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 79:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 80:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 81:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 82:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 83:

Using elementary transformations, find the inverse of each of the matrices, if it exists in

##### Answer:

##### Question 84:

Matrix A and B will be inverse of each other if

- AB = BA
- AB = BA = O
- AB = O, BA = I
- AB = BA = I.

##### Answer:

AB = BA = I.

##### Question 85:

If A = [a_{ij}]_{m x n} is a square matrix, if :

- m < n
- m > n
- m = n
- None of these.

##### Answer:

m = n

##### Question 86:

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

- Not possible to find

##### Answer:

Not possible to find

##### Question 87:

The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is :

- 27
- 18
- 81
- 512

##### Answer:

512

##### Question 88:

Assume X, Y, Z, W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3 and p x k respectively. Now answer the following

The restrictions on n, k and p so that PY + WY will be defined are :

- k = 3, p = n
- k is arbitrary, p = 2
- p is arbitrary
- k = 2, p = 3

##### Answer:

k = 3, p = n

##### Question 89:

Assume X, Y, Z, W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3 and p x k respectively. Now answer the following

If n = p, then the order of the matrix 7X - 5Z is :

- p x 2
- 2 x n
- n x 3
- p x n

##### Answer:

2 x n

##### Question 90:

If A, B are symmetric matrices of same order, then AB - BA is a :

- Skew-symmetric matrix
- Symmetric matrix
- Zero matrix
- Identity matrix.

##### Answer:

Skew-symmetric matrix

##### Question 91:

- $\mathrm{\pi}$

##### Answer:

##### Question 92:

Matrices A and B will be inverse of each other only if :

- AB = BA
- AB - BA = O
- AB = O, BA = I
- AB = BA = I

##### Answer:

AB = BA = I

##### Question 93:

- 1 + $\mathrm{\alpha}$
^{2}+ $\mathrm{\beta}$$\mathrm{\gamma}$ = 0 - 1 - $\mathrm{\alpha}$
^{2}+ $\mathrm{\beta}$$\mathrm{\gamma}$ = 0 - 1 - $\mathrm{\alpha}$
^{2}- $\mathrm{\beta}$$\mathrm{\gamma}$ = 0 - 1 + $\mathrm{\alpha}$
^{2}- $\mathrm{\beta}$$\mathrm{\gamma}$= 0

##### Answer:

1 - $\mathrm{\alpha}$^{2} - $\mathrm{\beta}$$\mathrm{\gamma}$ = 0

##### Question 94:

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

- A is a diagonal matrix
- A is a zero matrix
- A is a square matrix
- None of these.

##### Answer:

A is a zero matrix

##### Question 95:

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

- A
- I - A
- I
- 3A.

##### Answer:

I