Question 1

Evaluate the determinants in Exercises 1 and 2.

Question 2

Evaluate the determinants in Exercises 1 and 2.

(i) (ii)

Question 3

If, then show that

Question 4

If, then show that

Question 5

Evaluate the determinants

(i) (iii)

(ii) (iv)

Question 6

If, find.

Question 7

Find values of x, if

(i) (ii)

Question 8

If, then x is equal to

(A) 6 (B) ±6 (C) −6 (D) 0

Question 9

Which of the following is correct?

A. Determinant is a square matrix.

B. Determinant is a number associated to a matrix.

C. Determinant is a number associated to a square matrix.

D. None of these

Question 10

Prove that the determinant is independent of θ.

Question 11

Evaluate

Question 12

Let, where 0 ≤ θ≤ 2π, then

A. Det (A) = 0

B. Det (A) ∈ (2, ∞)

C. Det (A) ∈ (2, 4)

D. Det (A)∈ [2, 4]

Question 13

Using the property of determinants and without expanding, prove that:

Question 14

Using the property of determinants and without expanding, prove that:

Question 15

Using the property of determinants and without expanding, prove that:

Question 16

Using the property of determinants and without expanding, prove that:

Question 17

Using the property of determinants and without expanding, prove that:

Question 18

By using properties of determinants, show that:

(i)

(ii)

Question 19

By using properties of determinants, show that:

(i)

(ii)

Question 20

By using properties of determinants, show that:

Question 21

By using properties of determinants, show that:

Question 22

By using properties of determinants, show that:

(i)

(ii)

Question 23

By using properties of determinants, show that:

Question 24

By using properties of determinants, show that:

Question 25

By using properties of determinants, show that:

Question 26

By using properties of determinants, show that:

Question 27

Let A be a square matrix of order 3 × 3, then is equal to

A. B. C. D.

Question 28

Without expanding the determinant, prove that

Question 29

If a, b and c are real numbers, and,

Show that either a + b + c = 0 or a = b = c.

Question 30

Solve the equations

Question 31

Prove that

Question 32

Evaluate

Question 33

Using properties of determinants, prove that:

Question 34

Using properties of determinants, prove that:

Question 35

Using properties of determinants, prove that:

Question 36

Using properties of determinants, prove that:

Question 37

Using properties of determinants, prove that:

Question 38

Evaluate

Question 39

If a, b, c, are in A.P., then the determinant

A. 0 B. 1 C. x D. 2x

Question 40

Find area of the triangle with vertices at the point given in each of the following:

(i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8)

(iii) (−2, −3), (3, 2), (−1, −8)

Question 41

Show that points

are collinear

Question 42

Find values of k if area of triangle is 4 square units and vertices are

(i) (k, 0), (4, 0), (0, 2) (ii) (−2, 0), (0, 4), (0, k)

Question 43

(i) Find equation of line joining (1, 2) and (3, 6) using determinants

(ii) Find equation of line joining (3, 1) and (9, 3) using determinants

Question 44

If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is

A. 12 B. −2 C. −12, −2 D. 12, −2

Question 45

Write Minors and Cofactors of the elements of following determinants:

(i) (ii)

Question 46

(i) (ii)

Question 47

Using Cofactors of elements of second row, evaluate.

Question 48

Using Cofactors of elements of third column, evaluate

Question 49

If and Aij is Cofactors of aij, then value of Δ is given by

Question 50

Find adjoint of each of the matrices.

Question 51

Find adjoint of each of the matrices.

Question 52

Question 53

Question 54

Find the inverse of each of the matrices (if it exists).

.

Question 55

Find the inverse of each of the matrices (if it exists).

Question 56

Let and. Verify that

Question 57

If, show that. Hence find.

Question 58

For the matrix, find the numbers a and b such that A2 + aA + bI = O.

Question 59

For the matrixshow that A3 − 6A2 + 5A + 11 I = O. Hence, find A−1.

Question 60

If verify that A3 − 6A2 + 9A − 4I = O and hence find A−1

Question 61

Let A be a nonsingular square matrix of order 3 × 3. Then is equal to

A. B. C. D.

Question 62

If A is an invertible matrix of order 2, then det (A−1) is equal to

A. det (A) B. C. 1 D. 0

Question 63

Find the inverse of each of the matrices (if it exists).

Question 64

Find the inverse of each of the matrices (if it exists).

Question 65

Find the inverse of each of the matrices (if it exists).

Question 66

Find the inverse of each of the matrices (if it exists).

Question 67

Find the inverse of each of the matrices (if it exists).

Question 68

If

Question 69

Let verify that

(i)

(ii)

Question 70

If x, y, z are nonzero real numbers, then the inverse of matrix is

A. B.

C. D.

Question 71

Examine the consistency of the system of equations.

x + 2y = 2

2x + 3y = 3

Question 72

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

Question 73

Examine the consistency of the system of equations.

2x y = 5

x + y = 4

Question 74

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Question 75

Examine the consistency of the system of equations.

x + y + z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

Question 76

Examine the consistency of the system of equations.

3x y − 2z = 2

2yz = −1

3x − 5y = 3

Question 77

Examine the consistency of the system of equations.

5x y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

Question 78

Solve system of linear equations, using matrix method.

Question 79

Solve system of linear equations, using matrix method.

xy + z = 4

2x + y − 3z = 0

x + y + z = 2

Question 80

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3xy − 2z = 3

Question 81

Solve system of linear equations, using matrix method.

xy + 2z = 7

3x + 4y − 5z = −5

2x y + 3z = 12

Question 82

Solve system of linear equations, using matrix method.

Question 83

Solve system of linear equations, using matrix method.

Question 84

Solve system of linear equations, using matrix method.

Question 85

If, find A−1. Using A−1 solve the system of equations

Question 86

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg

wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70.

Find cost of each item per kg by matrix method.

Question 87

Solve the system of the following equations