NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 Determinants

Class 12 Maths Chapter 4 Exercise 4.3 Determinants:- In Chapter 4 of Class 12 Mathematics, we explore the important concept of determinants. Determinants are numerical values calculated from square matrices that help us understand various properties of these matrices. They are essential in many areas of mathematics and engineering.

Exercise 4.3 focuses on enhancing your understanding of determinants through different problems. You'll start by learning how to compute determinants for 2x2 and 3x3 matrices, which are the building blocks for larger matrices. This exercise will also introduce you to the ideas of minors and cofactors, which are smaller determinants used to find the determinants of larger matrices.

Additionally, Exercise 4.3 will cover several properties of determinants, such as how changing rows or columns affects their values. Get NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 Determinants from the below article. 

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NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.3 Determinants

Here are the Class 12 Maths Chapter 4 Exercise 4.3 Determinants NCERT Solutions below:- 

Question 1. Find the area of the triangle with vertices at the points 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)

Solution :

(i) The area of the triangle with vertices (1, 0), (6, 0), (4, 3) is given by the relation,

=

(ii) The area of the triangle with vertices (2, 7), (1, 1), (10, 8) is given by the relation,

=

(iii) The area of the triangle with vertices (−2, −3), (3, 2), (−1, −8)

=

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

Solution :

Therefore, points A, B and C are collinear.

Question 3. Find values of k if area of triangle is 4 sq. units and vertices are:

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

(ii) (−2, 0), (0, 4), (0, k)

Solution :

We know that the area of a triangle whose vertices are (x1, y1), (x2, y2), and

(x3, y3) is the absolute value of the determinant (Δ), where

When −k + 4 = − 4, k = 8.

When −k + 4 = 4, k = 0.

Hence, k = 0, 8.

(ii) The area of the triangle with vertices (−2, 0), (0, 4), (0, k) is given by the relation,

∴k − 4 = ± 4

When k − 4 = − 4, k = 0.

When k − 4 = 4, k = 8.

Hence, k = 0, 8.

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Question4. (i) Find the equation of the line joining (1, 2) and (3, 6) using determinants.

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

Solution :
(i) Let P(x, y) be any point on the line joining the points (1, 2) and (3, 6).

Then, Area of triangle that could be formed by these points is zero.

Hence, the equation of the line joining the given points is y = 2x.

(ii) Let P (x, y) be any point on the line joining points A (3, 1) and

B (9, 3). Then, the points A, B, and P are collinear. Therefore, the area of triangle ABP will be zero.

Hence, the equation of the line joining the given points is x − 3y = 0.

Question 5. 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

Solution :

The area of the triangle with vertices (2, −6), (5, 4), and (k, 4) is given by the relation,

It is given that the area of the triangle is ±35.

Therefore, we have:

⇒ 25 – 5k = ± 35

⇒ 5(5 – k) = ± 35

⇒ 5 – k = ± 7

When 5 − k = −7, k = 5 + 7 = 12.

When 5 − k = 7, k = 5 − 7 = −2.

Hence, k = 12, −2.

The correct answer is D.

Therefore, option (D) is correct.

Class 12 Maths Chapter 4 Exercise 4.3 Determinants Summary 

Check out: Class 12th Sample Papers

Basic Computation of Determinants

• Learn to compute the determinants of 2x2 and 3x3 matrices.

• Use the standard formulas to find the determinants, laying the foundation for understanding larger matrices.

Minors and Cofactors

• Understand the concepts of minors and cofactors, which are crucial for expanding the determinants of larger matrices.

• Practice calculating the cofactor matrix, an essential step in various matrix operations.

Properties of Determinants

• Explore key properties of determinants, such as how they change with row and column operations.

• Learn properties like the determinant of a transpose, the effect of row and column interchanges, and the determinant of a product of matrices.

Solving Linear Equations

• Apply determinants to solve systems of linear equations using Cramer’s Rule.

• This practical application highlights the utility of determinants in solving real-world problems.

Matrix Invertibility

• Use determinants to determine if a matrix is invertible.

• Understand that a non-zero determinant indicates an invertible matrix, while a zero determinant indicates a singular (non-invertible) matrix.

Check out: Class 12th Question Banks

Class 12 Maths Chapter 4 Exercise 4.3 Determinants  FAQs

Q1. What is a determinant?

Ans. A determinant is a numerical value calculated from a square matrix that provides information about the matrix, such as whether it is invertible and the volume scaling factor in linear transformations.

Q2. How do you find the determinant of a 2x2 matrix?

Ans. For a 2x2 matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}(ac​bd​), the determinant is calculated as ad−bcad - bcad−bc.

Q3. How do you find the determinant of a 3x3 matrix?

Ans. For a 3x3 matrix (abcdefghi)\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}​adg​beh​cfi​​, the determinant is calculated as a(ei−fh)−b(di−fg)+c(dh−eg)a(ei - fh) - b(di - fg) + c(dh - eg)a(ei−fh)−b(di−fg)+c(dh−eg).

Q4. What are minors and cofactors?

Ans. A minor of an element in a matrix is the determinant of the submatrix formed by deleting the row and column of that element. A cofactor is the minor multiplied by (−1)i+j(-1)^{i+j}(−1)i+j, where iii and jjj are the row and column indices of the element.

Q5. What properties of determinants are important?

Ans. Important properties include:

• The determinant of a matrix is zero if the matrix has two identical rows or columns.

• Swapping two rows or columns changes the sign of the determinant.

• The determinant of a product of matrices is the product of their determinants.

• The determinant of a transpose of a matrix is the same as the determinant of the original matrix.