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

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## Chapter 9 - Matrices Excercise Ex. 9(A)

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.

(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.

Given: , find x, y and z.

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

Solve for a, b and c if

(i)

(ii)

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

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

(i) A + B =

(ii) B - A

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

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

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

(i)B + C =

(ii)A - C =

(iii)A + B - C =

= =

(iv)A - B +C =

= =

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

(i)

(ii)

(iii)

(i)

(ii)

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

Find, x and y from the following equations :

(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.

Given: M =, find its transpose matrix M^{t}.
If possible, find:

(i) M + M^{t} (ii)
M^{t} - M

M =

M^{t} =

(i)

(i)

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

Where

We know additive inverse of a matrix is its negative.

Additive inverse of A =

Additive inverse of B =

Additive inverse of C =

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

(i) X + B = C - A

(ii) A - X = B + C

(i) X + B = C - A

(ii) A - X = B + C

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

(i) A + X = B

(ii) A - X = B

(iii) X - B = A

(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)

Evaluate:

Find x and y if:

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

Given ; find:

(i) 2A - 3B + C

(ii) A + 2C - B

(i) 2A - 3B + C

(ii) A + 2C - B

Given

(i) find the matrix 2A + B

(ii) find the matrix C such that:

C + B =

(i)

(ii) C + B =

C = - B =

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

Comparing the corresponding elements, we get,

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

3y = 15 y = 5

z = 9

Given A = and A^{t}
is its transpose matrix. Find:

(i) 2A + 3A^{t} (ii)
2A^{t} - 3A

(iii) (iv)

(i) 2A + 3A^{t}

(ii) 2A^{t} - 3A

(iii)

(iv)

Given

Solve for matrix X:

(i) X + 2A = B

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

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

(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

If , show that:

3M + 5N =

3M + 5N

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

(i) M - 2I =

(ii) 5M + 3I =

(i) M - 2I =

(ii) 5M + 3I =

If

2M =

M =

## Chapter 9 - Matrices Excercise Ex. 9(C)

Evaluate: if possible:

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.

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

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

(iv) IB (v) A^{2} (vi)
B^{2}A

Find x and y, if:

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

If , find:

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

Is A(BC) = (AB)C?

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

Given , find; if possible:

(i) AB (ii) BA (iii)A^{2}

(iii) Product AA (=A^{2}) is not
possible as the number of columns of matrix A is not equal to its number of
rows.

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

M^{2} = 2M + 3I

Hence, M^{2} = 2M + 3I.

If and BA= M^{2},
find the values of a and b.

Given,
BA = M^{2}

Comparing the corresponding elements, we get,

a = 2

-2b = -2 b = 1

Given , find:

(i) A - B (ii) A^{2}

(iii) AB (iv) A^{2}
- AB + 2B

If ; find:

(i) (A + B)^{2} (ii) A^{2}
+ B^{2}

(iii) Is (A + B)^{2} = A^{2} + B^{2}?

(iii)
Clearly, (A + B)^{2} A^{2}
+ B^{2}

Find the matrix A, if B = and B^{2}
= B + A.

B^{2}
= B + A

A = B^{2}
- B

A
= 2(B^{2} - B)

If A = and A^{2} = I; find a and b.

It is given that A^{2} = I.

Comparing the corresponding elements, we get,

1 + a = 1

Therefore, a = 0

-1 + b = 0

Therfore, b = 1

If ; then show that:

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

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

If , simplify:

A^{2} + BC.

Solve for x and y:

Solve for x and y:

In each case given below, find:

(a) The order of matrix M.

(b) The matrix M.

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

If ; find the
value of x, given that: A^{2} = B.

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?

AB = BA = B

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

Hence, A is an identity matrix.

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

Comparing the corresponding elements, we get,

3a = 3 + a

2a = 3

a =

3b = b b = 0

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

If , then compute:

(i) P^{2} - Q^{2} (ii)
(P + Q) (P - Q)

Is (P + Q) (P - Q) = P^{2} - Q^{2}
true for matrix algebra?

Clearly, it can be said that:

(P
+ Q) (P - Q) = P^{2} - Q^{2} not true for matrix algebra.

Given the matrices:

. Find:

(i) ABC (ii) ACB.

State whether ABC = ACB.

Hence, ABC ≠ ACB.

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

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

Thus, CA + B A + CB

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

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

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

If , find:

(i) A^{t}. A (ii) A. A^{t}

Where A^{t} is the transpose of matrix A.

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

Hence, proved.

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

Comparing the corresponding elements, we get,

2x + 12 = 0

thus, x = -6

6 + 6y = 0

thus, y = -1

Evaluate without using tables:

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) A^{2} - B^{2} = (A + B) (A - B)

(viii) (A - B)^{2} = A^{2} - 2A. B + B^{2}

(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)

Find x and y, if:

Comparing the corresponding elements, we get,

6x - 10 = 8

6x = 18

x = 3

-2x + 14 = 4y

4y = -6+ 14 = 8

y = 2

Find x and y, if:

Comparing the corresponding elements, we get,

3x + 18 = 15

3x = -3

x = -1

12x + 77 = 10y

10y = -12 + 77 = 65

y = 6.5

If ; find x and y, if:

(i) x, y Î W (whole numbers)

(ii) x, y Î Z (integers)

(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.

(i)

(ii)

Evaluate:

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

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

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

Comparing the corresponding elements, we get,

a + 1 = 5 a = 4

2 + b = 0 b = -2

-1 - c = 3 c = -4

If A = ; find:

(i) A (BA)

(ii) (AB). B

(i)

(ii)

Find x and y, if:

Comparing the corresponding elements, we get,

5x = 5x = 1

6y = 12 y = 2

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

Given ; find the matrix X such that:

A + X = 2B + C

Given, A + X = 2B + C

Find the value of x, given that A^{2} = B,

Given, A^{2} = B

Comparing the corresponding elements, we get,

x = 36

If , and I is identity matrix of the same order and A^{t} is the transpose of matrix A, find A^{t }.B + BI

Let. Find A^{2} - A + BC.

Let A =. Find A^{2} + AB + B^{2}.

A =

A^{2} = A A =

=

AB = A B =

=

=

B^{2} = B x B =

=

=

A^{2} + AB + B^{2} =

=

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

Comparing the corresponding elements, we get,

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

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

11 = 6c c =

Given A =.

Find the values of p and q.

A =

BA =

C^{2} =

BA = C^{2} =

By comparing,

-2q = -8 q = 4

And p = 8

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

AB =

Evaluate:

=

=

A^{2} =
9A + MI

⇒ A^{2}
- 9A = mI ….(1)

Now, A^{2}
= AA

Substituting A^{2} in (1), we have

A^{2} - 9A = mI

(i) Write the order of matrix X.

(ii) Find the matrix 'X'

(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

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

Given: A^{2}
- 5B^{2} = 5C

Given matrix . Find the matrix X if, X = B^{2} - 4B. Hence,
solve for a and b given .

To find: a and b

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