#### Exercise 4.1

**Question 1. Evaluate the following determinant :**

**Question 2. **

**Question 3. **

**Question 4.**

**Question 5. Evaluate the determinants :**

**Question 6.**

**Question 7. Find the value of x, if**

**Question 8.**

#### Exercise 4.2

**Using the property of determinants and without expanding in Questions 1 to 5, prove that**

**Question 1. **

**Question 2. **

**Question 3. **

**Question 4. **

**Question 5. **

**By using properties of determinants, in Questions 6 to 14, show that**

**Question 6.**

**Question 7.**

**Question 8.**

**Question 9.**

**Question 10.**

**Note:** In case of determinants, we can use both elementary row operations and elementary column operations to arrive at the result, as you have seen in the above question.

**Question 11.**

**Question 12.**

**Question 13.**

**Question 14.**

**Question 15. If A be a square matrix of order 3 × 3, then | kA | is equal to**

**Question 16. 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**

**Sol. (c) **Since we know that, determinant of any matrix is defined if and only if Ais a square matrix.\ determinant is a number associated to a square matrix.

#### Exercise 4.3

**Question 1. Find the 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)**

**Note: **Since area is positive, we always use the absolute value of the determinant (Which is used for the area).

**Question 2. Show that the points A (a, b + c) , B (b, c + a) C (c, a + b) are collinear**

**Hint: **Use area of triangle

**Note: **Area of a triangle formed by three collinear points is zero.

**Question 3. Find the value 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)**

**Note: **If area of a triangle is given, we use both positive and negative values of the determinant for calculations.

**Question 4. (i) Find the equation of line joining (1, 2) and (3, 6) using determinants.**

**(ii) Find the equation of line joining (3, 1), (9, 3) using determinants**.

**Question 5. If area of triangle is 35 sq. units with vertices (2, – 6), (5, 4) and (k, 4). Then k is (a) 12 (b) – 2 (c) – 12, – 2 (d) 12, – 2**

#### Exercise 4.4

**Question 1. Write the minors and cofactors of the elements of following determinants :**

**Question 2. Write Minors and Cofactor of elements of the determinant**

**Question 3. Using cofactors of elements of second row, evaluate Del =**

**Question 4. Using Cofactors of elements of third column, evaluate **

**Question 5. **

#### Exercise 4.5

**Question. Find the adjoint of each of the matrices in Exercises 1 and 2**.

**Question. Verify A (adj A) = (adj A) × A = |A| I in Questions 3 and 4**

**Question . Find the inverse of each of the matrices (if it exists) given in Questions 5 to 11 :**

**Question 12. **

**Question 13. **

**Question 14. **

**Question 15. **

**Question 16. **

**Question 17. Let A be a non-singular square matrix of order 3 × 3. Then | Adj A | is equal to :**

**Question 18. If A is an invertible matrix of order 2, then det. (A ^{–1}) is equal to :**

#### Exercise 4.6

**Question. Examine the consistency of the system of equations in Questions 1 to 6 :**

**Question : Solve the following of linear equations using matrix method in Exercises 7 to 14 :**

**Question 15. **

**Question 16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is rs. 69. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is rs. 90. The cost of 6 kg onion, 2 kg wheatand 3 kg rice is rs.70. Find the cost of each item per kg by matrix method.**

#### Miscellaneous Exercise

**Question 1. Prove that the determinant**

**Question 2. Without expanding the determinant, prove that**

**Question 3. Evaluate**

**Question 4.**

**Question 5.** **Solve the equation**

**Question 6.**

**Question 7.**

**Question 8.**

**Question 9.**

**Question 10. Evaluate**

**Using properties of determinants in exercises 11 to 15, prove that :**

**Question 11. **

**Question 12. **

**Question 13. **

**Question 14. **

**Question 15. **

**Question 16. Solve the system of the following equations**

**Question . Choose the correct answer in Questions 17 to 19**

**Question 18. If x, y, z are nonzero real numbers, then the inverse of matrix**

**Question 19.**

**Related Articles:**

- NCERT Solutions for Class 12 (All Subjects)
- NCERT Solutions for Class 12 Maths (Chapter-wise PDF)
- Determinants Class 12 Notes Mathematics Chapter 4