ICSE-X-Mathematics
08: Matrices Class 10 Maths ML Aggarwal Solutions
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 get3 = 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. If
find 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. If
is 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. If
find 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. If
Find 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 get8x = 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 elements2y = 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 elementsa = 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 elements3a + 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
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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 elements3x = 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 elements6 + 6y = 0
⇒ 6y = - 6
⇒ y = - 1
2x + 12 = 0
⇒ 2x = - 12
⇒ x = - 6
Hence x = - 6, y = - 1
34. Let
where 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 elementsx = 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) If
find 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 get3a - 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. If
Find 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. If
where 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 getx + 3 = 5
⇒ x = 5 - 3 = 2
And y - 4 = 3
⇒ y = 3 + 4 = 7
x = 2, y = 7
3. If
then 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 get3x = 6
⇒ x = 6/3 = 2
⇒ - y = 3
⇒ x = 2, y = - 3
4. If
then 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. If
then 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. If
then 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. If
then the matrix A is equal to
(a)
(b)
(c)
(d)
Answer
(d)
Given:
8. If
then 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. If
then 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 elements2a - 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)
