RD Sharma Solution CLass 12 Mathematics Chapter 7 Adjoint and Inverse of a Matrix

Chapter 7 – Adjoint and Inverse of a Matrix Exercise Ex. 7.1

Question 1
Solution 1
Question 2

Solution 2

Question 3
Solution 3

Question 4
Solution 4

Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24

Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32

Solution 32

Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37

Question 38
Solution 38
Question 39
Solution 39
Question 40
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43

Show that A3 – 6A2 + 5A + 11I3 = Ο. Hence, find A-1.

Solution 43

Question 44
Solution 44
Question 45

Verify that A3 – 6A2 + 9A – 4I = Ο and hence find A-1.

Solution 45

Question 46
Solution 46
Question 47
Solution 47
Question 48
Solution 48
Question 49
Solution 49
Question 50
Solution 50
Question 51
Solution 51
Question 52
Solution 52
Question 53
Solution 53
Question 54
Solution 54
Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7

Question 8
Solution 8

Question 9
Solution 9

Question 10
Solution 10
Question 11
Solution 11

Question 12
Solution 12

Question 13
Solution 13

Question 14
Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Chapter 7 – Adjoint and Inverse of a Matrix Exercise MCQ

Question 1

If A is an invertible matrix, then which of the following is not true

a. (A2)-1 = (A-1)2

b. |A-1| = |A|-1

c. (AT)-1 = (A-1)T

d. |A| ≠ 0

Solution 1

Correct option: (a)

|A-1| = |A|-1, (AT)-1 = (A-1)T, |A| ≠ 0 are properties of an invertible matrix.

Question 2

If A is an invertible matrix of order 3, then which of the following is not true

b. (A-1)-1 = A

c. If BA = CA, then B ≠ C, where B and C are square matrices of order 3

d. (AB)-1 = B-1 A-1, where B = [bij]3×3 and |B| ≠ 0

Solution 2

Correct option: (c)

Question 3

a. is a skew-symmetric matrix

b. A-1 + B-1

c. Does not exist

d. None of these

Solution 3

Correct option:(d)

Question 4

a.

b.

c.

d.

Solution 4

Correct option: (b)

Question 5

If A is singular matrix, then adj A is

a. non-singular

b. singular

c. symmetric

d. not defined

Solution 5

Correct option:(b)

If A is singular matrix then adjoint of A is also singular.

Question 6

If A, B are two n × n non – singular matrices, then

a. AB is non-singular

b. AB is singular

c. (AB)-1 = A-1B-1

d. (AB)-1 does not exist

Solution 6

Correct option: (a)

Question 7

a. a27

b. a9

c. a6

d. a2

Solution 7

Correct option: (c)

Question 8

a. 144

b. 143

c. 142

d. 14

Solution 8

Correct option:(a)

Question 9

If B is non-singular matrix and A is a square matrix, then det (B-1AB) is equal to

a. Det (A-1)

b. Det (B-1)

c. Det (A)

d. Det (B)

Solution 9

Correct option: (c)

Question 10

a. 20

b. 100

c. 10

d. 0

Solution 10

Correct option: (c)

Question 11

If A5 = O such that An≠ I for 1 ≤ n ≤ 4, then (I – A)-1 equals

a. A4

b. A3

c. I + A

d. none of these

Solution 11

Correct option: (d)

Question 12

If A satisfies the equation x3 – 5x2 + 4x + λ = 0, then A-1 exists if

a. λ ≠ 1

b. λ ≠ 2

c. λ ≠ -1

d. λ ≠ 0

Solution 12

Correct option: (d)

Question 13

If for the matrix A, A3 = I, then A-1 =

a. A2

b. A3

c. A

d. none of these

Solution 13

Correct option: (a)

Question 14

If A and B are square matrices such that B = – A– 1 BA, then (A + B)2 =

a. O

b. A2 + B2

c. A2 + 2AB + B2

d. A + B

Solution 14

Correct option: (b)

Question 15

a. 5A

b. 10A

c. 16A

d. 32A

Solution 15

Correct option: (c)

Question 16

For non-singular square matrix A,B and C of the same order (AB-1C)-1 =

a. A-1BC-1

b. C-1 B-1A-1

c. CBA-1

d. C-1BA-1

Solution 16

Correct option: (d)

Question 17

a. -3

b. 3

c. 0

d. Non-existent

Solution 17

Correct option: (d)

Question 18

If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is

a. dn

b. dn-1

c. dn+1

d. d

Solution 18

Correct option: (b)

Question 19

If A is a matrix of order 3 and |A| = 8, then |adj A| =

a. 1

b. 2

c. 23

d. 26

Solution 19

Correct option: (d)

Question 20

If A2 – A + I = O, then the inverse of A is

a. A-2

b. A + I

c. I – A

d. A – I

Solution 20

Correct option: (c)

Question 21

If A and B are invertible matrices, which of the following statement is not correct.

a. Adj A = |A| A-1

b. Det(A-1) = (det A)-1

c. (A + B)-1 = A-1 + B-1

d. (AB)-1 = B-1A-1

Solution 21

Correct option: (c)

Adj A = |A| A-1, Det(A-1) = (det A)-1, (AB)-1 = B-1A-1 are all the properties of invertible matrix.

Question 22

If A is a square matrix such that A2 = I, then A-1 is equal to

a. A + I

b. A

c. 0

d. 2A

Solution 22

Correct option: (b)

Question 23

a.

b.

c.

d. none of these

Solution 23

Correct option: (a)

Question 24

a. 19

b. 1/19

c. -19

d. -1/19

Solution 24

Correct option: (b)

Question 25

a. 3

b. 0

c. -3

d. 1

Solution 25

Question 26

a. A

b. -A

c. ab A

d. none of these

Solution 26

Correct option: (d)

Question 27

a. a = 1, b = 1

b. a = cos 2θ, b = sin 2θ

c. a = sin 2θ, b = cos

d. none of these

Solution 27

Correct option: (b)

Question 28

If a matrix A is such that 3A3 + 2A2 + 5A + I = 0, then A-1 is equal to

a. – (3A2 + 2A + 5)

b. 3A2 + 2A + 5

c. 3A2 – 2A – 5

d. none of these

Solution 28

Correct option: (d)

Question 29

If A is an invertible matrix, then det (A-1) is equal to

a. det (A)

b.

c. 1

d. none of these

Solution 29

Correct option: (b)

Question 30

a.

b.

c.

d. none of these

Solution 30

Correct option: (a)

Question 31

If x, y, z are non-zero real numbers, then the inverse of the matrix

a.

b.

c.

d.

Solution 31

Correct option: (a)

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