RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 7. Adjoint and Inverse of a Matrix

Subject: Maths - Class 12th

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Chapter Exercises

Exercise 7.1
Exercise 7.2

Exercise 7.1

1

Find the adjoint of each of the following Matrices.


Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
1

Find the adjoint of each of the following Matrices.


Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
1

Find the adjoint of each of the following Matrices.


Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
1

Find the adjoint of each of the following Matrices.


Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
2

Find the adjoint of each of the following Matrices and Verify that (adj A) A = |A| I = A (adj A)


Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
2

Find the adjoint of each of the following Matrices and Verify that (adj A) A = |A| I = A (adj A)


Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
2

Find the adjoint of each of the following Matrices and Verify that (adj A) A = |A| I = A (adj A)


Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
2

Find the adjoint of each of the following Matrices and Verify that (adj A) A = |A| I = A (adj A)


Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
3

For the matrix A= , show that A(adj A)=O.
4

If , show that adj A=A.
5

If , show that adj A=3AT.
6

Find A (adj A) for the matrix A=.
7

Find the inverse of each of the following matrices:
7

Find the inverse of each of the following matrices:
7

Find the inverse of each of the following matrices:
7

Find the inverse of each of the following matrices:
8

Find the inverse of each of the following matrices.
8

Find the inverse of each of the following matrices.
8

Find the inverse of each of the following matrices.
8

Find the inverse of each of the following matrices.
8

Find the inverse of each of the following matrices.
8

Find the inverse of each of the following matrices.
8

Find the inverse of each of the following matrices.
9

Find the inverse of each of the following matrices and verify that A – 1 A = I3.
9

Find the inverse of each of the following matrices and verify that A – 1 A = I3.
10

For the following pairs of matrices verify that (AB)–1 = B – 1A – 1:

A=
10

For the following pairs of matrices verify that (AB)–1 = B – 1A – 1:
11

Let . Find (AB) – 1.
12

Given , compute A – 1 and show that 2A – 1 = 9I – A.
13

If , then show that A – 3I = 2 (I + 3A – 1).
14

Find the inverse of the matrix and show that aA – 1 = (a2 + bc + 1) I – aA.
15

Given . Compute (AB) – 1.
16

Let and . Show that

[F (α)] – 1 = F( – α)
16

Let and . Show that

[G(β)] – 1 = G( – β)
16

Let and . Show that

[F(α)G(β)] – 1 = G – ( – β) F( – α).
17

If , verify that A2 – 4 A + I = O, where and . Hence, find A – 1.
18

Show that satisfies the equation A2 + 4A – 42I = O. Hence, find A – 1.
19

If , show that A2 – 5A + 7I = O. Hence, find A – 1.
20

If A = find x and y such A2 – xA + yI = O. Hence, evaluate A – 1.
21

If , find the value of λ so that A2 = λA – 2I. Hence, find A – 1.
22

Show that satisfies the equation x2 – 3A – 7 = 0. Thus, find A – 1.
23

Show that satisfies the equation x2–12 x + 1 = 0. Thus, find A – 1
24

For the matrix . Show that A3 – 6A2 + 5A + 11I3 = O.Hence, find A – 1.
25

Show that the matrix, satisfies the equation, A3 – A2 – 3A – I3 = O. Hence, find A–1.
26

If . Verify that A3 – 6A2 + 9A – 4I = O and hence fid A – 1.
27

If , prove that A – 1 = AT.
28

If , show that A – 1 = A3.
29

If , Show that A2 = A–1.
30

Solve the matrix equation , where X is a 2x2 matrix.
31

Find the matrix X satisfying the matrix equation: .
32

Find the matrix X for which: .
33

Find the matrix X satisfying the equation: .
34

If , find A–1 and prove that A2 – 4A–5I = O.
35

If A is a square matrix of order n, prove that |A adj A| = |A|n.
36

If A – 1 = and , find (AB) – 1.
37

If , find (AT) – 1.
38

Find the adjoint of the matrix and hence show that A(adj A) = |A| I3.
39

If , find A – 1 and show that A – 1 = 1/2(A2 – 3I).

Exercise 7.2

1

Find the inverse of each of the following matrices by using elementary row transformations:
2

Find the inverse of each of the following matrices by using elementary row transformations:
3

Find the inverse of each of the following matrices by using elementary row transformations:
4

Find the inverse of each of the following matrices by using elementary row transformations:
5

Find the inverse of each of the following matrices by using elementary row transformations:
6

Find the inverse of each of the following matrices by using elementary row transformations:
7

Find the inverse of each of the following matrices by using elementary row transformations:
8

Find the inverse of each of the following matrices by using elementary row transformations:
9

Find the inverse of each of the following matrices by using elementary row transformations:
10

Find the inverse of each of the following matrices by using elementary row transformations:
11

Find the inverse of each of the following matrices by using elementary row transformations:
12

Find the inverse of each of the following matrices by using elementary row transformations:
13

Find the inverse of each of the following matrices by using elementary row transformations:
14

Find the inverse of each of the following matrices by using elementary row transformations:
15

Find the inverse of each of the following matrices by using elementary row transformations:
16

Find the inverse of each of the following matrices by using elementary row transformations: