# ML Aggarwal Solutions for Chapter 8 Matrices Class 10 Maths ICSE

Here, we are providing the solutions for Chapter 8 Matrices from ML Aggarwal Textbook for Class 10 ICSE Mathematics. Solutions of the eighth chapter has been provided in detail. This will help the students in understanding the chapter more clearly. Class 10 Chapter 8 Matrices ML Aggarwal Solutions for ICSE is one of the most important chapter for the board exams which is based on classifying the matrices, addition, subtraction and multiplication of matrices, finding the value of variables of equation using matrix and also construction of matrices.

### Exercise 8.1

1. Classify the following matrices:

(i)

(ii)

(iii)
(iv)
(v)
(vi)
Answer

(i) It is square matrix of order 2

(ii) It is column matrix of order 1 × 3

(iii) It is column matrix of order 3 × 1

(iv) It is matrix of order 3 × 2

(v) It is matrix of order 2 × 3

(vi) It is zero matrix or order 2 × 3

2. (i) If a matrix has 4 elements, what are the possible order it can have ?

(ii) If a matrix has 8 elements, what are the possible order it can have ?

Answer

(i) It can have 1 × 4, 4 × 1, or 2 × 2 order

(ii) It can have 1 × 8, 8 × 1, 2 × 4 or 4 × 2 order

3. Construct a 2×2 matrix whose elements aij are given by

(a) aij = 2i – j

(b) aij = i, j

Answer:

(i) It can be

(ii) It can be

4. Find the values of x and y if :

Answer

Comparing corresponding elements,

2x + y = 5 ...(i)

3x – 2y = 4 …(ii)

Multiply (i) by 2 and (ii) by ‘1’ we get

4x + 2y = 10, 3x – 2y = 4

Adding we get, 7x = 14

⇒ x = 2

Substituting the value of x in (i)

2 × 2 + y = 5

⇒ 4 + y = 5

y = 5 – 4 = 1

Hence x = 2, y = 1

5. Find the value of x if  =

Answer

Comparing the corresponding terms, we get,

- y = 2

⇒ y = - 2

3x + y = 1 ⇒ 3x ≠ 1 – y

⇒ 3x = 1 – (-2)

= 1 + 2

= 3

⇒ x = 3/3 = 1

Hence x = 1, y = - 2

6. If  =find values of x and y

Answer

Comparing the corresponding terms, we get,

x + 3 = 5

⇒ x = 5 – 3 = 2

⇒ y – 4 = 3

⇒ y = 3 + 4 = 7

x = 2, y = 7

7. Find the values of x, y and z if

Answer

Comparing the corresponding elements of equal determinants,

x + 2 = - 5

⇒ x = - 5 – 2 = - 7

∴ x = - 7, 5z = - 20

⇒ z = - 20/5 = - 4

⇒ y2 + y = 6

⇒ y2 + y – 6 = 0

⇒ y2 + 3y - 2y – 6 = 0

⇒ y(y + 3) – 2(y+ 3) = 0

⇒ (y + 3)(y – 2) = 0

Either y + 3 = 0, then y = - 3 or y – 2 = 0, then y = 2

Hence x = - 7, y = - 3, 2, z = - 4

8. Find the values of x, y, a and b if

Answer

Comparing corresponding elements

x – 2 = 3, y = 1

x = 3 + 2 = 5

a + 2b = 5 ….(i)

3a – b = 1 ....(ii)

Multiplying (i) by 1 and (ii) by 2

a + 2b = 5, 6a – 2b = 2

Adding, we get, 7a = 7

⇒ a = 1

Substituting the value of a in (i)

1 + 2b = 5

⇒ 2b = 5 – 1 = 4

⇒ b = 2

Hence x = 5, y = 1, a = 1, b = 2

9. Find the values of a, b, c and d if  =

Answer

Comparing the corresponding terms, we get

3 = d

⇒ d = 3

⇒ 5 + c = - 1

⇒ c = - 1 – 5

⇒ c = - 6

a + b = 6 and ab = 8

∴ (a – b)2 = (a + b)2 – 4ab

= (6)2 – 4 × 8 = 36 – 32

= 4

= (± 2)2

∴ a - b = ± 2

(i) If a – b = 2

a + b = 6

Adding, we get 2a = 8 ⇒ a = 4

a + b = 6

⇒ 4 + b = 6

⇒ b = 6 – 4 = 2

(ii) If a – b = - 2

a + b = 6

Adding, we get, 2a = 4

⇒ a = 4/2 = 2

a + b = 6

⇒ 2 + b = 6

⇒ b = 6 – 2 = 4

∴ a = 2, b = 4

Hence, a = 4, b = 2, or a = 2, b = 4

c = - 6 and d = 3

10. Find the values of x, y and b, if

Answer

Comparing the corresponding terms, we get,

3x + 4y = 2 …..(i)

x – 2y = 4 …(ii)

Multiplying (i) by 1 and (ii) by 2

3x + 4y = 2, 2x – 4y = 8

Adding we get, 5x = 10

⇒ x = 2

Substituting the value of x in (i)

3 × 2 + 4y = 2,

6 + 4y = 2,

4y = 2 – 6 = - 4

y = - 1

∴ x = 2, y = - 1

a + b = 5 …(iii)

2a – b = - 5 …(iv)

### Exercise 8.2

1. Given that

find M + 2N

Answer

2. If

Find 2A – 3B

Answer

3. If

Compute 3A + 4B

Answer

4. Given

(i) Find the matrix 2A + B

(ii) Find a matrix 2A + B

Answer

5.

Find A + 2B – 3C

Answer

6. If

Find the matrix X if:

(i) 3A + X = B

(ii) X – 3B = 2A

Answer

(i) 3A + X = B

⇒ X = B – 3A

(ii) X – 3B = 2A

⇒ X = 2A + 3B

7. Solve the matrix equation

Answer:

8. If

find the matrix M

Answer

9. Given

Find the matrix X such that A + 2X = 2B + C

Answer

A + 2X = 2B + C

2X = 2B + C – A

10. Find X and Y if X + Y =

Answer

11. Iffind the values of x and y

Answer

12. If

Find the values of x and y

Answer

13. If

Find the values of x and y.

Answer

Comparing the corresponding terms, we get

4 – 4x = - 8

⇒ - 4x = - 8 – 4

⇒ - 4x = - 12

⇒ x = (- 12/-4) = 3

And y + 5 = 2

⇒ y = 2 – 5 = - 3

∴ x = 3, y = - 3

14.

Find the value of a, b and c.

Answer

Comparing the corresponding elements:

a + 1 = 5 ⇒ a = 4

b + 2 = 0 ⇒ b = - 2

- c = 3 ⇒ c = - 3

15. If

and 5A + 2B = C, Find the values of a, b, c.

Answer

Comparing each term

5a + 6 = 9

⇒ 5a = 9 – 6 = 3

⇒ a = 3/5

⇒ 25 + 2b = - 11

⇒ 2b = - 11 – 25 = - 36

⇒ b = - (36/2) = - 18

c = 6

Hence a = 3/5, b = - 18 and c = 6

### Exercise 8.3

1. Ifis the product AB possible ? Give a reason. If yes, find AB.

Answer

Yes, the product is possible because of number of column in A = number of row in B

i.e., (2 × 2) . (2 × 1) = (2 × 1) is the order of the matrix.

2. Iffind AB and BA, Is AB = BA ?

Answer

3. If

Find 2PQ.

Answer

4. Given A =evaluate A2 – 4A

Answer:

5. If Find AB – 5C.

Answer

6. If find A(BA)

Answer

7. Given the matrices:

Find the products of (i) ABC (ii) ACB and state whether they are equal.

Answer

8. Evaluate:

Answer:

9. If

find the matrix AB + BA

Answer

10. If
Find each of the following and state if they are equal.

(i) CA + B

(ii) A + CB

Answer

(i) CA + B

(ii)

We can say that CA + B ≠ A + CB

11. If
Find 2B – A2

Answer

12. If
Compute:

(i) A(B + C)

(ii) (B + C)A

Answer

13. IfFind the matrix C(B – A)

Answer

14. Let
Find A2 + AB + B2

Answer

Given that

15. Let
Find A2 + AC – 5B

Answer

16. If A =

find A2 and A3. Also state that which of these is equal to A.

Answer

From above, it is clear that A3 = A

17. If X =

Show that 6X – X2 = 9I where I is the unit matrix.

Answer

Given that

= 9I = R.H.S

Hence, proved.

18. Show that is a solution of the matrix equation X2 – 2X – 3I = 0, where I is the unit matrix of order 2

Answer

Given, X2 – 2X – 3I = 0

∴ X2 – 2X – 3I = 0

Hence proved.

19. Find the matrix X or order 2 × 2 which satisfies the equation+ 2X =

Answer

20. If A =find the value of x, so that A2 – 0

Answer

Comparing 1+ x = 0 ⇒ x = - 1

21. If  =Find the value of x

Answer

Comparing the corresponding elements

x= -1

22. (i) Find x and y if

(ii) Find x and y if  =

Answer

Comparing the corresponding elements

- 3x + 4 = - 5

⇒ - 3x = - 5 – 4 = - 9

- 10 = y

⇒ y = - 10

Hence x = 3, y = - 10

(ii)

Comparing, we get

8x = 16

⇒ x = 16/8 = 2

And 9y = 9

⇒ y = 9/9 = 1

Here x = 2, y = 1

23. Find x and y if  =

Answer

Comparing the corresponding elements

2x + y = 3 …(i)

3x + y = 2 ....(ii)

Subtracting, we get

- x = 1

⇒ x = - 1

Substituting the value of x in (i)

2(-1) + y = 3

⇒ - 2 + y = 3

⇒ y = 3 + 2 = 5

Hence, x = - 1, y = 5

24. If  =find the values of x and y

Answer

Comparing the corresponding elements

2y = 0 ⇒ y = 0

3x = 9 ⇒ x = 3

Hence x = 3, y = 0

Question 25: If

Write down the values of a, b, c and d

Answer

Comparing the corresponding elements

a = 3, b = 4, c = 2, d = 5

26. Find the value of x given that A2 = B

Answer

A2 = B

⇒ A × A = B

Comparing the corresponding elements of two equal matrices, x = 36.

27. If A =

find the value of x, given that A2 – B.

Answer

Corresponding the corresponding elements 3x = 36

⇒ x = 12

Hence x = 12

28. If

find x and y when A2 = B

Answer

⇒ 4x = 16 and 1 = - y

⇒ x = 4 and y = - 1

29. Find x, y if

Answer

⇒ 2x = 6 and 2y = - 4

⇒ x = 6/2 and y = - 4/2

⇒ x = 3 and ⇒ y = - 2

30.If  find a, b and c

Answer

Comparing the corresponding elements

3a + 2 = 11

⇒ 3a = 11 – 2 = 9

∴ a = 9/3 = 3

4a – 3 = b

⇒ b = 4 × 3 – 3

= 12 - 3 = 9

⇒ 3 = c

Hence a= 3, b = 9, c = 3

31. If find the value of x if AB = BA

Answer

Comparing the corresponding elements

x – 2 = 8 – x

⇒ x + x = 8 + 2

⇒ 2x = 10

∴ x = 10/2

= 5

32. If A = find x and y so that A2 – xA + yI

Answer

Comparing the corresponding elements

3x = 12

⇒ x = 4

2x + y = 7

⇒ 2 ×4 + y = 7

⇒ 8 + y = 7

⇒ y = 7 – 8 = - 1

Hence x = 4, y = - 1

33. If

Find x and y such that PQ = 0

Answer

Comparing the corresponding elements

6 + 6y = 0

⇒ 6y = - 6

⇒ y = - 1

2x + 12 = 0

⇒ 2x = - 12

⇒ x = - 6

Hence x = - 6, y = - 1

34. Letwhere M is a matrix

(i) State the order of matrix M

(ii) Find the matrix M

Answer

Given (i) M is the order of 1 × 2

Comparing the corresponding elements

x = 1 and x + 2y = 2

⇒ 1 + 2y = 2

⇒ 2y = 2 – 1 = 1

⇒ y = 1/2

Hence x = 1, y = 1/2

35. Given , X =

(i) the order of matrix X

(ii) the matrix X

Answer

2x + y = 7 …(i)

- 3x + 4y = 6 ...(ii)

Multiplying (i) by 3 and (ii) by 2, and adding we get:

6x + 3y = 21

- 6x + 8y = 12

11y = 33 ⇒ y = 3

From (i), 2x = 7 – 3 = 4

⇒ x = 2

36. Solve the matrix equation:

Answer

Comparing the corresponding elements.

4x = - 4 ⇒ x = - 1

4y = 8 ⇒ y = 2

37. (i) Iffind the matrix C such that AC = B.

(ii) If  find the matrix C such that CA = B.

Answer

(i)

Comparing the corresponding elements,

2x – y = - 3 …(i)

-4x + 5y = 2 ….(ii)

Multiplying (i) by 5 and (ii) by 1

10x – 5y = - 15

- 4x + 5y = 2

Adding, we get 6x = - 13

⇒ x = - 13/6

Substituting the value of x in (i)

2(-13/6) – y = - 3

⇒ - 13/3 – y = - 3

⇒ - y = 3 + 13/3

= (-9 + 13)/3

= 4/3

∴ y = - (4/3)

(ii)

Comparing,

2x – 4y = 0

⇒ x – 2y = 0

∴ x = 2y

And –x + 5y = - 3

⇒ - 2y + 5y = - 3

⇒ 3y = - 3

⇒ y = - 1

∴ x = 2y = 2 × (-1) = - 2

38. If A =find matrix B such that BA = I, where I is unity matrix of order 2.

Answer

Comparing the corresponding terms, we get

3a – b = 1, - 4a + 2b = 0

⇒ 2b = 4a

⇒ b = 2a

∴ 3a – b = 1

⇒ 3a – 2a = 1

⇒ a = 1

and b = 2a

⇒ b = 2 × 1 = 2

∴ a = 1, b = 2

and 3c – d = 0 ⇒ d = 3c

- 4c + 2d = 1

⇒ - 4c + 2 × 3c = 1

⇒ - 4c + 6c = 1

⇒ 2c = 1

⇒ c = 1/2

And d = 3c = 3 × 1/2 = 3/2

Hence a = 1, b = 2, c = 1/2, d = 3/2

39. IfFind the matrix A such that AB = C

Answer

Comparing corresponding elements, we get

∵ - 4a + 5b = 17 ...(i)

2a – b = - 1 …(ii)

- 4c + 5d = 47 ...(iii)

2c – d = - 13 …(iv)

Multiplying (i) by 1 and (ii) by 2

⇒ - 4a + 5b = 17

4a – 2b = - 2

Adding 3b = 15

⇒ b = 15/3 = 5

2a – b = - 1

⇒ 2a – 5 = - 1

⇒ 2a = - 1 + 5 = 4

⇒ a = 4/2 = 2

∴ a = 2, b = 5

Again multiplying(iii) by 1 and (iv) by 2,

- 4c + 5d = 47

4c – 2d = - 26

Adding 3d = 21

⇒ d = 21/3 = 7

And 2c – d = - 13

⇒ 2c – 7 = - 13

⇒ 2c = - 13 + 7 = - 6

⇒ c = - 6/2 = - 3

∴ c = - 3, d = 7

### Multiple Choice Questions

Choose the correct answer from the given four options (1 to 14):

1. Ifwhere aij = i + j, then A is equal to

(a)

(b)

(c)

(d)

Answer

(b)

A = 2×2 where aij = i + j, then A is equal to

2. If then the values of x and y are

(a) x = 2, y = 7

(b) x = 7, y = 2

(c) x = 3, y = 6

(d) x = - 2, y = 7

Answer

(d) x = - 2, y = 7

Comparing we get

x + 3 = 5

⇒ x = 5 – 3 = 2

And y – 4 = 3

⇒ y = 3 + 4 = 7

x = 2, y = 7

3. Ifthen the values of x and y are

(a) x = 2, y = 3

(b) x = 2, y = - 3

(c) x = - 2,

(d) x = 3, y = 2

Answer

(b) x = 2, y = - 3

Comparing, we get

3x = 6

⇒ x = 6/3 = 2

⇒ - y = 3

⇒ x = 2, y = - 3

4. Ifthen the value of x is

(a) - 2

(b) 0

(c) 2

(d) 2

Answer

(d) 2

Comparing, we get

y = - 2

And x – 2y = 6

⇒ x – 2 × (-2) = 6

⇒ x + 4 = 6

⇒ x = 6 – 4 = 2

5. Ifthen the value of x – y is

(a) – 3

(b) 1

(c) 3

(d) 5

Answer

(c) 3

Comparing, we get

3y = - 3

⇒ y = -3/3 = - 1

4x = 8

⇒ x = 8/4 = 2

x – y = 2 – (-1)

= 2 + 1 = 3

6. Ifthen the values of x and y are

(i) x = 2, y = 6

(b) x = 2, y = -6

(c) x = 3, y = - 4

(d) x = 3, y = - 6

Answer

(b) x = 2, y = - 6
Comparing, we get

3x = 6

⇒ x = 6/3 = 2

And 3x – y = 10

2 ×2 – y = 10

⇒ 4 – y = 10

⇒ - y = 10 – 4 = 6

⇒ y = - 6

∴ x = 2, y = -6

7. Ifthen the matrix A is equal to

(a)

(b)

(c)

(d)

Answer

(d)

Given:

8. Ifthen A2 is equal to

(a)

(b)

(c)

(d)

Answer

(c)

9. A = then A2 =

(a)

(b)

(c)

(d)

Answer

(c)

10. If A = then A2 =

(a)

(b)

(c)

(d)

Answer

Given

11. If A =then A2 =

(a) A

(b) 0

(c) I

(d) 2A

Answer

(b) 0

Given:

12. If A =then A2

(a)

(b)

(c)

(d) none of these

Answer

(c)

Given,

13. Ifthen A2 =

(a)

(b)

(c)

(d)

Answer

(a)

14. If A = , then A2 = pA, then the value of p is

(a) 2

(b) 4

(c) – 2

(d) – 4

Answer

(b) 4

Comparing, we get

8 = 2p

p = 4

### Chapter Test

1. Find the values of a and b if

Answer

Comparing the corresponding elements

a + 3 = 2a + 1

⇒ 2a – a = 3 – 1

⇒ a = 2

b2 + 2 = 3b

⇒ b2 – 3b + 2 = 0

⇒ b2 – b – 2b + 2 = 0

⇒ b(b – 1) – 2(b – 1) = 0

(b – 1)(b – 2) = 0

Either b – 1 = 0, then b = 1

Or b -2 = 0, then b = 1

Or b – 2 = 0, then b = 2

Hence, a = 2, b = 2 or 1

2. Find a, b, c and d if

Answer

Comparing the corresponding elements:

3a = 4 + a

⇒ 3a – a = 4

⇒ 2a = 4

∴ a = 2

3b = a + b + 6

⇒ 3b – b = 2 + 6

⇒ 2b = 8

∴ b = 4

3d = 3 + 2d ⇒ 3d – 2d = 3

∴ d = 3

3c = c + d – 1

⇒ 3c – c = 3 – 1

2c = 2

⇒ c = 1

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

3. Find X if

Answer

Given

4. Determine the matrices A and B when

Answer

5. (i) Find the matrix B if and A2 = A + 2B

(ii) If
And C =find A(4B – 3C)

Answer

(i)

Comparing the corresponding elements

4 + 2a = 18

⇒ 2a = 18 – 4 = 14

∴ a = 7

1 + 2b = 7

⇒ 2b = 7 – 1 = 6

∴ b = 3

2 + 2c = 14

⇒ 2c = 14 – 2 = 12

∴ c = 6

3 + 2d = 11

⇒ 2d = 11 – 3 = 8

∴ d = 4

Hence a = 7, b = 3, c = 6, d = 4

(ii)

6. If

Compute (AB)C = (CB) A ?

Answer

Given

It is clear from above that (AB)C ≠ (CB)A.

7. If  find each of the following and state if they are equal:

(i) (A + B)(A – B)

(ii) A2 – B2

Answer

Given

(ii)

We see that (A + B)(A – B) ≠ A2 – B2

8. If A = find A2 – 5A – 14I

Where I is unit of order 2 × 2

Answer

9. If A =and A2 = 0 find p and q

Answer

Comparing the corresponding elements

9 + 3p = 0

⇒3p = - 9

⇒ p = - 3

9 + 3q = 0

⇒ 3q = - 9

⇒ q = - 3

Hence p = -3, q = - 3

10. If A =

and A2 = I, find x, y

Answer

Given

Comparing the corresponding elements,

9/25 + 2/5.x = 1

⇒ 2/5.x = 1 = 9/25 = 16/25

x = 16/25 × 5/2 = 8/5

6/25 + 2/5.y = 0

⇒ 2/5y = -6/25

y = - 6/25 × 5/2 = -3/5

Hence x = 8/5, y = -3/5

11. If find a, b, c and d

Answer

Comparing the corresponding elements

- a = 1

⇒ a = - 1

-b = 0

⇒ b = 0

c = 0 and d = - 1

Hence a = - 1, b = 0, c = 0, d = - 1

12. Find a and b if

Answer

Comparing the corresponding elements

2a – 4 = 0

⇒ 2a = 4

⇒ a = 2

2a – 2b = - 2

⇒ 2 × 2 – 2b = - 2

⇒ 4 – 2b = - 2

⇒ - 2b = - 2 – 4

= - 6

⇒ b = 3

Hence a = 2, b = 3

13. If

Find (i) 2A – 3B

(ii) A2

(iii) BA

Answer:

Given

(∵ cot 45° = 1)

(i) 2A – 3B

(ii) A2 = A × A

(iii)

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