RS Aggarwal - Mathematics

Book: RS Aggarwal - Mathematics

Chapter: 5. Matrices

Subject: Maths - Class 12th

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Chapter Exercises

Exercise 5A
Exercise 5B
Exercise 5C
Exercise 5D
Exercise 5E
Exercise 5F

Exercise 5A

1

If then write

i. the number of rows in A,


ii. the number of columns in A,


iii. the order of the matrix A,


iv. the number of all entries in A,


v. the elements a23, a31, a14, a33, a22 of A.
2

Write the order of each of the following matrices:

i.


ii.


iii.


iv. D = [8 -3]


v.


vi, F = [6]
3

If a matrix has 18 elements, what are the possible orders it can have?
4

Find all possible orders of matrices having 7 elements.
5

Construct a 3 × 2 matrix whose elements are given by aij = (2i – j).
6

Construct a 4 × 3 matrix whose elements are given by

7

Construct a 2 × 2 matrix whose elements are

8

Construct a 2 × 3 matrix whose elements are

9

Construct a 3 × 4 matrix whose elements are given by

Exercise 5B

1

If and verify that (A + B) = (B + A).
2

If and verify that (A + B) + C = A + (B+C).
3

If and find (2A – B).
4

Let and Find:

i. A + 2B


ii. B – 4c


iii. A – 2B + 3C
5

Let and Compute 5A – 3B + 4C.
6

If find A.
7

Find matrices A and B, if and

8

Find matrices A and B, if and

9

Find matrix X, if

10

If and find a matrix C such that A + B – C = O.
11

Find the matrix X such that 2A – B + X = O,

where and


12

If and find a matrix C such that (A + B + C) is a zero matrix.
13

If A = diag [2, -5, 9], B = diag [-3, 7, 14] and C = diag [4, -6, 3], find:

(i) A + 2B


(ii)B + C – A
14

Find the value of x and y, when

i.


15

Find the value of (x + y) from the following equation :
16

If then write the value of (x + y).

Exercise 5C

1 A

Compute AB and BA, which ever exists when

and


1 B

Compute AB and BA, which ever exists when

and


1 C

Compute AB and BA, which ever exists when

and


1 D

Compute AB and BA, which ever exists when

A = [1 2 3 4] and


1 E

Compute AB and BA, which ever exists when

and


2 A

Show that AB ≠ BA in each of the following cases :

and


2 B

Show that AB ≠ BA in each of the following cases :

and


3 A

Show that AB = BA in each of the following cases:

and


3 B

Show that AB = BA in each of the following cases:

and


3 C

Show that AB = BA in each of the following cases:

and


4

If and shown that AB = A and BA = B.
5

If and , show that AB is a zero matrix.
16 A

For the following matrices, verify that A(BC) = (AB)C :

and


6

For the following matrices, verify that A(BC) = (AB)C :

and C = [1 -2]
7 A

Verify that A(B + C) = (AB + AC), when

and


7 B

Verify that A(B + C) = (AB + AC), when

and


8

If and verify that A(B – C) = (AB – AC).
9

If show that A2 = O.
10

If show that A2 = A.
11

If show that A2 = I.
12

If and find (3A2 – 2B + I).
13

If then find (-A2 + 6A).
14

If show that (A2 – 5A + 7I) = O.
15

Show that the matrix satisfies the equation A3 – 4A2 + A = O.
16

If find k so that A2 = kA – 2I.
17

If find f(A), where f(x) = x2 – 2x + 3.
18

If and f(x) = 2x3 + 4x + 5, find f(A).
19

Find the values of x and y, when
20

Solve for x and y, when
21

If find x and y such that A2 + xI = yA.
22

If find the value of a and b such that A2 + aA + bI = O.
23

Find the matrix A such that

24

Find the matrix A such that A.

25

If and (A + B)2 = (A2 + B2) then find the values of a and b.
26

If show that F(x) . F(y) = F(x + y).
27

If show that

28

If find x.
29

If find x.
30

Find the values of a and b for which
31

If find f(A), where f(x) = x2 – 5x + 7.
32

If prove that for all n ∈ N.
33

Given an example of two matrices A and B such that

A ≠ O, B ≠ O, AB = O and BA ≠ O.
34

Give an example of three matrices A, B, C such that

AB = AC but B ≠ C.
35

If and find (3A2 – 2B + I).
36

If find the value of x.

Exercise 5D

1

If verify that (A’)’ = A.
2

If verify that (2A)’ = 2A’.
3

If and verify that (A + B)’ = (A’ + B’).
4

If and verify that (P + Q)’ = (P’ + Q’).
5

If show that (A + A’) is symmetric.
6

If show that (A + A’) is skew-symmetric.
7

Show that the matrix is skew-symmetric.

HINT: Show that A’ = -A.
8

Express the matrix as the sum of a symmetric matrix and a skew-symmetric matrix.
9

Express the matrix as the sum of a symmetric matrix and a skew-symmetric matrix.
10

Express the matrix as the sum of a symmetric and a skew-symmetric matrix.
11

Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where

12

Express the matrix as sum of two matrices such that one is symmetric and the other is skew-symmetric.
13 A

For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :

and


13 B

For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :

and


13

For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :

and B = [-2 -1 -4]
13 D

For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :

and


14

If show that A’A = I.
15

If matrix A = [1 2 3], write AA’.

Exercise 5E

1

Using elementary row transformations, find the inverse of each of the following matrices:
2

Using elementary row transformations, find the inverse of each of the following matrices:
3

Using elementary row transformations, find the inverse of each of the following matrices:
4

Using elementary row transformations, find the inverse of each of the following matrices:
5

Using elementary row transformations, find the inverse of each of the following matrices:
6

Using elementary row transformations, find the inverse of each of the following matrices:
7

Using elementary row transformations, find the inverse of each of the following matrices:
8

Using elementary row transformations, find the inverse of each of the following matrices:
9

Using elementary row transformations, find the inverse of each of the following matrices:
10

Using elementary row transformations, find the inverse of each of the following matrices:
11

Using elementary row transformations, find the inverse of each of the following matrices:
12

Using elementary row transformations, find the inverse of each of the following matrices:
13

Using elementary row transformations, find the inverse of each of the following matrices:
14

Using elementary row transformations, find the inverse of each of the following matrices:
15

Using elementary row transformations, find the inverse of each of the following matrices:

Exercise 5F

1

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

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

If find the values of x and y.
4

Find the values of x and y, if
5

If find the values of x and y.
6

If find the values of x, y, z, ω.
7

If find the values of x, y, z, ω.
8

If A = diag (3 -2, 5) and B = diag (1 3 -4), find (A + B).
9

Show that
10

If and find the matrix C such that A + B + C is a zero matrix
11

If then find the least value of α for which A + A’ = I.
12

Find the value of x and y for which
13

Find the value of x and y for which
14

If show that (A + A’) is symmetric
15

If and show that (A – A’) is skew-symmetric
16

If and find a matrix X such that A + 2B + X = O.
17

If and find a matrix X such that

3 A – 2B + X = O.
18

If show that A’ A = I.
19

If A and B are symmetric matrices of the same order, show that (AB – BA) is a skew symmetric matrix.
20

If and f(x) = x2 – 4x + 1, find f(A).
21

If the matrix A is both symmetric and skew-symmetric, show that A is a zero matrix.