Chapter 9 : Matrices - Selina Solutions for Class 10 Maths ICSE

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Chapter 9 - Matrices Excercise Ex. 9(A)

Question 1

State, whether the following statements are true or false. If false, give a reason.

(i) If A and B are two matrices of orders 3 2 and 2 3 respectively; then their sum A + B is possible.

(ii) The matrices and are conformable for subtraction.

(iii) Transpose of a 2 1 matrix is a 2 1 matrix.

(iv) Transpose of a square matrix is a square matrix.

(v) A column matrix has many columns and one row.

Solution 1

(i) False

The sum A + B is possible when the order of both the matrices A and B are same.

(ii) True

(iii) False

Transpose of a 2 1 matrix is a 1 2 matrix.

(iv) True

(v) False

A column matrix has only one column and many rows.

Question 2

Given: , find x, y and z.

Solution 2

If two matrices are equal, then their corresponding elements are also equal. Therefore, we have:

x = 3,

y + 2 = 1 y = -1

z - 1 = 2 z = 3

Question 3

Solve for a, b and c if

(i)

(ii)

Solution 3

If two matrices are equal, then their corresponding elements are also equal.

(i)

a + 5 = 2 a = -3

-4 = b + 4 b = -8

2 = c - 1 c = 3

(ii) a= 3

a - b = -1

b = a + 1 = 4

b + c = 2

c = 2 - b = 2 - 4 = -2

Question 4

If A = and B = ; find: (i) A + B (ii) B - A

Solution 4

(i) A + B =

(ii) B - A

Question 5

If A=, B = and C = ; find:

(i) B + C (ii) A - C

(iii) A + B - C (iv) A - B +C

Solution 5

(i)B + C =

(ii)A - C =

(iii)A + B - C =

= =

(iv)A - B +C =

= =

Question 6

Wherever possible, write each of the following as a single matrix.

(i)

(ii)

(iii)

Solution 6

(i)

 

(ii)

 

(iii) Addition is not possible, because both matrices are not of same order.

Question 7

Find, x and y from the following equations :

Solution 7

(i)

Equating the corresponding elements, we get,

3 - x = 7 and y + 2 = 2

Thus, we get, x = - 4 and y = 0.

(ii)

Equating the corresponding elements, we get,

-8 + y = -3 and x - 2 =2

Thus, we get, x = 4 and y = 5.

Question 8

Given: M =, find its transpose matrix Mt. If possible, find:

(i) M + Mt (ii) Mt - M

Solution 8

M =

Mt =

(i)

(i)

Question 9

Write the additive inverse of matrices A, B and C:

Where

Solution 9

We know additive inverse of a matrix is its negative.

Additive inverse of A =

Additive inverse of B =

Additive inverse of C =

Question 10

Given ; find the matrix X in each of the following:

(i) X + B = C - A

(ii) A - X = B + C

Solution 10

(i) X + B = C - A


(ii) A - X = B + C

Question 11

Given ; find the matrix X in each of the following:

(i) A + X = B

(ii) A - X = B

(iii) X - B = A

Solution 11

(i) A + X = B

X = B - A


 

(ii) A - X = B

X = A - B

 


 

(iii) X - B = A

X = A + B

Chapter 9 - Matrices Excercise Ex. 9(B)

Question 1

Evaluate:

Solution 1

Question 2

Find x and y if:

Solution 2

Comparing the corresponding elements, we get,

12 + 2y = 10 and 3x - 6 = 0

Simplifying, we get, y = -1 and x = 2.

Comparing corresponding the elements, we get,

-x + 8 = 7 and 2x - 4y = -8

Simplifying, we get,

x = 1 and y = = 2.5

Question 3

Given ; find:

(i) 2A - 3B + C

(ii) A + 2C - B

Solution 3

(i) 2A - 3B + C

(ii) A + 2C - B

Question 4

Solution 4

Question 5

Given

(i) find the matrix 2A + B

(ii) find the matrix C such that:

C + B =

Solution 5

(i)

(ii) C + B =

C = - B =

Question 6

If ; find the values of x, y and z.

Solution 6

Comparing the corresponding elements, we get,

2x + 9 = -7 2x = -16 x = -8

3y = 15 y = 5

z = 9

Question 7

Given A = and At is its transpose matrix. Find:

(i) 2A + 3At (ii) 2At - 3A

(iii) (iv)

Solution 7

(i) 2A + 3At

(ii) 2At - 3A

(iii)

(iv)

Question 8

Given

Solve for matrix X:

(i) X + 2A = B

(ii) 3X + B + 2A = O

(iii) 3A - 2X = X - 2B.

Solution 8

(i) X + 2A = B

X = B - 2A

(ii) 3X + B + 2A = O

3X = -2A - B

(iii) 3A - 2X = X - 2B

3A + 2B = X + 2X

3X = 3A + 2B

Question 9

If , show that:

3M + 5N =

Solution 9

3M + 5N

Question 10

If I is the unit matrix of order 2 x 2; find the matrix M, such that:

(i) M - 2I =

(ii) 5M + 3I =

Solution 10

(i) M - 2I =

(ii) 5M + 3I =

Question 11

If

Solution 11

2M =

M =

Chapter 9 - Matrices Excercise Ex. 9(C)

Question 1

Evaluate: if possible:

Solution 1

The number of columns in the first matrix is not equal to the number of rows in the second matrix. Thus, the product is not possible.

Question 2

If and I is a unit matrix of order 2 2, find:

(i) AB (ii) BA (iii) AI

(iv) IB (v) A2 (vi) B2A

Solution 2

Question 3

Solution 3

 

Question 4

Find x and y, if:

Solution 4

Comparing the corresponding elements, we get,

5x - 2 = 8 x = 2

20 + 3x = y y = 20 + 6 = 26

Comparing the corresponding elements, we get,

x = 2

-3 + y = -2 y = 1

Question 5

If , find:

(i) (AB)C (ii) A(BC)

Is A(BC) = (AB)C?

Solution 5

Hence, A(BC) = (AB)C.

Question 6

Given , find; if possible:

(i) AB (ii) BA (iii)A2

Solution 6

(iii) Product AA (=A2) is not possible as the number of columns of matrix A is not equal to its number of rows.

Question 7

Solution 7

 

Question 8

If M = and I is a unit matrix of the same order as that of M; show that:

M2 = 2M + 3I

Solution 8

 

Hence, M2 = 2M + 3I.


Question 9

If and BA= M2, find the values of a and b.

Solution 9

Given, BA = M2

Comparing the corresponding elements, we get,

a = 2

-2b = -2 b = 1

Question 10

Given , find:

(i) A - B (ii) A2

(iii) AB (iv) A2 - AB + 2B

Solution 10

Question 11

If ; find:

(i) (A + B)2 (ii) A2 + B2

(iii) Is (A + B)2 = A2 + B2?

Solution 11

(iii) Clearly, (A + B)2 A2 + B2

Question 12

Find the matrix A, if B = and B2 = B + A.

Solution 12

B2 = B + A

A = B2 - B

A = 2(B2 - B)

Question 13

If A = and A2 = I; find a and b.

Solution 13

It is given that A2 = I.

 

Comparing the corresponding elements, we get,

1 + a = 1

Therefore, a = 0

 

 

 

-1 + b = 0 

Therfore, b = 1

Question 14

If ; then show that:

(i) A (B + C) = AB + AC

(ii) (B - A)C = BC - AC.

Solution 14

Question 15

If , simplify:

A2 + BC.

Solution 15

Question 16

Solve for x and y:

Solution 16

Question 17

Solve for x and y:

Solution 17

Question 18

Solution 18

 

Question 19

In each case given below, find:

(a) The order of matrix M.

(b) The matrix M.

 

Solution 19

We know, the product of two matrices is defined only when the number of columns of first matrix is equal to the number of rows of the second matrix.

 

(i) Let the order of matrix M be a x b.

Clearly, the order of matrix M is 1 x 2.

 

Comparing the corresponding elements, we get,

a = 1 and a + 2b = 2 2b = 2 - 1 = 1 b =

 

(ii) Let the order of matrix M be a x b.

Clearly, the order of matrix M is 2 x 1.

 

Comparing the corresponding elements, we get,

a + 4b = 13 ....(1)

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

 

Multiplying (2) by 4, we get,

8a + 4b = 20 ....(3)

 

Subtracting (1) from (3), we get,

7a = 7 a = 1

From (2), we get,

b = 5 - 2a = 5 - 2 = 3

Question 20

If ; find the value of x, given that: A2 = B.

Solution 20

Question 21

Solution 21

Question 22

If A and B are any two 2 x 2 matrices such that AB = BA = B and B is not a zero matrix, what can you say about the matrix A?

Solution 22

AB = BA = B

We know that I × B = B × I = B, where I is the identity matrix.

Hence, A is an identity matrix.

Question 23

Given and that AB = A + B; find the values of a, b and c.

Solution 23

Comparing the corresponding elements, we get,

3a = 3 + a

2a = 3 

a =            

 

 

3b = b b = 0

4c = 4 + c 3c = 4 c =

Question 24

If , then compute:

(i) P2 - Q2 (ii) (P + Q) (P - Q)

Is (P + Q) (P - Q) = P2 - Q2 true for matrix algebra?

Solution 24

Clearly, it can be said that:

(P + Q) (P - Q) = P2 - Q2 not true for matrix algebra.

Question 25

Given the matrices:

. Find:

(i) ABC (ii) ACB.

State whether ABC = ACB.

Solution 25

 

 

Hence, ABC ≠ ACB.

Question 26

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

(i) CA + B (ii) A + CB

Solution 26

 

Thus, CA + B   A + CB


Question 27

If ; find the matrix X such that AX = B.

Solution 27

Clearly, the order of matrix X is 2 x 1.

 

Comparing the two matrices, we get,

2x + y = 3 … (1)

x + 3y = -11 … (2)

Multiplying (1) with 3, we get,

6x + 3y = 9 … (3)

Subtracting (2) from (3), we get,

5x = 20

x = 4

From (1), we have:

y = 3 - 2x = 3 - 8 = -5

 

Question 28

If, find (A - 2I) (A - 3I).

Solution 28

Question 29

If , find:

(i) At. A (ii) A. At

Where At is the transpose of matrix A.

Solution 29

Question 30

If, show that: 6M - M2 = 9I; where I is a 2 x 2 unit matrix.

Solution 30

Hence, proved.


Question 31

If; find x and y such that PQ = null matrix.

Solution 31

 

Comparing the corresponding elements, we get,

2x + 12 = 0

thus, x = -6

6 + 6y = 0

thus, y = -1

Question 32

Evaluate without using tables:

Solution 32

Question 33

State, with reason, whether the following are true or false. A, B and C are matrices of order 2 x 2.

(i) A + B = B + A

(ii) A - B = B - A

(iii) (B. C). A = B. (C. A)

(iv) (A + B). C = A. C + B. C

(v) A. (B - C) = A. B - A. C

(vi) (A - B). C = A. C - B. C

(vii) A2 - B2 = (A + B) (A - B)

(viii) (A - B)2 = A2 - 2A. B + B2

Solution 33

(i) True.

Addition of matrices is commutative.

(ii) False.

Subtraction of matrices is not commutative.

(iii) True.

Multiplication of matrices is associative.

(iv) True.

Multiplication of matrices is distributive over addition.

(v) True.

Multiplication of matrices is distributive over subtraction.

(vi) True.

Multiplication of matrices is distributive over subtraction.

(vii) False.

Laws of algebra for factorization and expansion are not applicable to matrices.

(viii) False.

Laws of algebra for factorization and expansion are not applicable to matrices.

Chapter 9 - Matrices Excercise Ex. 9(D)

Question 1

Find x and y, if:

Solution 1

Comparing the corresponding elements, we get,

6x - 10 = 8

6x = 18

x = 3

-2x + 14 = 4y

4y = -6+ 14 = 8

y = 2

Question 2

Find x and y, if:

Solution 2

Comparing the corresponding elements, we get,

3x + 18 = 15

3x = -3

x = -1

12x + 77 = 10y

10y = -12 + 77 = 65

y = 6.5

Question 3

If ; find x and y, if:

(i) x, y Î W (whole numbers)

(ii) x, y Î Z (integers)

Solution 3

(i) x, y Î W (whole numbers)

It can be observed that the above two equations are satisfied when x = 3 and y = 4.

 

(ii) x, y Î Z (integers)

It can be observed that the above two equations are satisfied when x = 3 and y = 4.

Question 4

 

 

 

Solution 4

(i)

 

 

 

(ii)

 

 

 

 

Question 5

Evaluate:

Solution 5

Question 6

If and 3A x M = 2B; find matrix M.

Solution 6

Let the order of matrix M be a x b.

 

3A x M = 2B

Clearly, the order of matrix M is 2 x 1.


Comparing the corresponding elements, we get,

-3y = -10


 y =           

12x - 9y = 12

 

 

Question 7

If , find the values of a, b and c.

Solution 7

 

Comparing the corresponding elements, we get,

                                 

a + 1 = 5 a = 4

2 + b = 0 b = -2

-1 - c = 3 c = -4

Question 8

If A = ; find:

(i) A (BA)

(ii) (AB). B

Solution 8

(i)

(ii)

Question 9

Find x and y, if:

Solution 9

Comparing the corresponding elements, we get,

5x = 5x = 1

6y = 12 y = 2

Question 10

If matrix X = and 2X - 3Y =; find the matrix 'X' and 'Y'.

Solution 10

Question 11

Given ; find the matrix X such that:

A + X = 2B + C

Solution 11

Given, A + X = 2B + C


Question 12

Find the value of x, given that A2 = B,

Solution 12

 

Given, A2 = B


Comparing the corresponding elements, we get,

x = 36

Question 13

If , and I is identity matrix of the same order and At is the transpose of matrix A, find At .B + BI

Solution 13

Question 14

 

Solution 14

Question 15

Let. Find A2 - A + BC.

Solution 15

Question 16

Let A =. Find A2 + AB + B2.

Solution 16

A =

A2 = A A =

=

AB = A B =

=

=

B2 = B x B =

=

=

A2 + AB + B2 =

=

Question 17

If and 3A - 2C = 6B, find the values of a, b and c.

Solution 17

 

Comparing the corresponding elements, we get,

3a - 8 = 24 3a = 32 a =

24 - 2b = 0 2b = 24 b = 12

11 = 6c c =

Question 18

Given A =.

Find the values of p and q.

Solution 18

A =

BA =

C2 =

BA = C2 =

By comparing,

-2q = -8 q = 4

And p = 8

Question 19

Given A = . Find AB + 2C - 4D.

Solution 19

AB =

Question 20

Evaluate:

Solution 20

=

=

Question 21

 

 

Solution 21

 

 

 

 

 

 

 

 

Question 22

Solution 22

A2 = 9A + MI

⇒ A2 - 9A = mI ….(1)

Now, A2 = AA

Substituting A2 in (1), we have

A2 - 9A = mI

 

Question 23

(i) Write the order of matrix X.

(ii) Find the matrix 'X'

Solution 23

 

(i) Let the order of matrix X = m × n

Order of matrix A = 2 × 2

Order of matrix B = 2 × 1

Now, AX = B

  

∴ m = 2 and n = 1

 

Thus, order of matrix X = m × n = 2 × 1

 

Multiplying (1) by 2, we get

4x + 2y = 8 ….(3)

Subtracting (2) from (3), we get

3x = 3

⇒ x = 1

Substituting the value of x in (1), we get

2(1) + y = 4

⇒ 2 + y = 4

⇒ y = 2

Question 24

Find the matrix C where C is a 2 by 2 matrix. 

Solution 24

Given: A2 - 5B2 = 5C

Question 25

Given matrix . Find the matrix X if, X = B2 - 4B. Hence, solve for a and b given .

Solution 25

To find: a and b

 

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