**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**

Choose the correct answer.

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**

Choose the correct answer.

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**

Choose the correct answer.

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

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

**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 A_{ij} is Cofactors of *a*_{ij}, 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**

Verify *A* (*adj A*) = (*adj A*) *A* = *I* .

**Question 53**

Verify *A* (*adj A*) = (*adj A*) *A* = *I* .

**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 *A*^{2} + *aA* + *bI *= *O*.

**Question 59**

For the matrixshow that *A*^{3} − 6*A*^{2} + 5*A* + 11 *I* = O. Hence, find *A*^{−1.}

**Question 60**

If verify that *A*^{3} − 6*A*^{2} + 9*A* − 4*I* = *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**

Choose the correct answer.

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 *+ 2*y *= 2

2*x* + 3*y *= 3

**Question 72**

Solve system of linear equations, using matrix method.

5*x* + 2*y* = 3

3*x* + 2*y* = 5

**Question 73**

Examine the consistency of the system of equations.

2*x *− *y* = 5

*x* + *y *= 4

**Question 74**

Examine the consistency of the system of equations.

*x* + 3*y* = 5

2*x* + 6*y* = 8

**Question 75**

Examine the consistency of the system of equations.

*x* +* y *+ *z* = 1

2*x* + 3*y* + 2*z* = 2

*ax* + *ay* + 2*az* = 4

**Question 76**

Examine the consistency of the system of equations.

3*x* −* y *− 2z = 2

2*y* − *z* = −1

3*x* − 5*y* = 3

**Question 77**

Examine the consistency of the system of equations.

5*x* −* y *+ 4*z* = 5

2*x* + 3*y* + 5*z* = 2

5*x* − 2*y* + 6*z* = −1

**Question 78**

Solve system of linear equations, using matrix method.

**Question 79**

Solve system of linear equations, using matrix method.

*x* − *y* + *z* = 4

2*x* + *y* − 3*z* = 0

*x* + *y* + *z* = 2

**Question 80**

Solve system of linear equations, using matrix method.

2*x* + 3*y* + 3*z* = 5

*x* − 2*y* + *z* = −4

3*x* − *y* − 2*z* = 3

**Question 81**

Solve system of linear equations, using matrix method.

*x* − *y* + 2*z* = 7

3*x* + 4*y* − 5*z* = −5

2*x* −* y* + 3*z* = 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

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