NCERT Solutions for Class 12 Maths Chapter 4: Determinants

Introduction

The National Council of Educational Research and Training (NCERT) textbooks play an essential role in shaping the foundation of mathematical knowledge for students across India. Chapter 4 of the Class 12 NCERT Mathematics book, titled Determinants, is a vital topic within the curriculum. This chapter extends the concepts introduced in Chapter 3 (Matrices) and introduces students to the importance of determinants in solving systems of linear equations, matrix algebra, and various practical applications.

In this article, we’ll provide a comprehensive overview of Chapter 4, highlight the importance of understanding determinants, and explore solutions to the exercises from this chapter, based on the NCERT Class 12 Mathematics textbook. Our discussion will also help students grasp the theory, solve problems effectively, and score well in the board examinations.

Overview of Determinants

What is a Determinant?

A determinant is a scalar value that can be computed from the elements of a square matrix. It is a unique number associated with a square matrix and provides critical insights into various properties of the matrix, including invertibility, linear dependence, and more. Determinants are primarily used to solve systems of linear equations, find the area of parallelograms and triangles, and have applications in advanced fields such as computer graphics and physics.

Key Topics Covered in Chapter 4

Definition of a Determinant
The chapter begins by introducing the definition of determinants and their connection to matrices. A determinant is defined only for square matrices (i.e., matrices with the same number of rows and columns), and the value of the determinant depends on the elements of the matrix.
Order of Determinants
The determinant is calculated based on the order of the matrix. In this chapter, students learn how to compute determinants for 2x2 and 3x3 matrices. For example, the determinant of a 2x2 matrix $\left[\begin{matrix} a & b \\ c & d \end{matrix}\right]$ is given by $ad - bc$ .
Properties of Determinants
One of the most critical parts of this chapter is understanding the properties of determinants, such as linearity, product rule, and the effect of row/column operations. These properties simplify determinant computation and are useful in matrix transformations.
Applications of Determinants
Determinants have practical applications, particularly in solving systems of linear equations using methods such as Cramer’s Rule. They also help in understanding whether a matrix is invertible.
Cofactors and Minors
Cofactors and minors are essential for expanding determinants of larger matrices (like 3x3 and beyond). The minor of an element is the determinant of the matrix formed by deleting the row and column of that element, and the cofactor is the minor multiplied by a sign factor.
Adjoint and Inverse of a Matrix
Students learn how to calculate the adjoint of a matrix (the transpose of the cofactor matrix) and use this to find the inverse of a matrix. The inverse of a matrix exists only if its determinant is non-zero.
Applications in Geometry
Determinants are used to calculate the area of triangles and parallelograms when their vertices are known. They also have implications in calculating volumes in higher dimensions.

Detailed NCERT Solutions for Class 12 Maths Chapter 4: Determinants

Let’s walk through some of the key exercises and their NCERT solutions from Chapter 4. This will help students understand the application of the above-mentioned concepts.

Exercise 4.1: Determinants of 2x2 and 3x3 Matrices

Question Example:
Find the determinant of the matrix:

A = \begin{bmatrix} 3 & 8 \\ 4 & 6 \end{bmatrix}

Solution:
The determinant of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is calculated as:

|A| = ad - bc

For the given matrix $A = \begin{bmatrix} 3 & 8 \\ 4 & 6 \end{bmatrix}$ , we apply the formula:

|A| = (3 \times 6) - (8 \times 4) = 18 - 32 = -14

Thus, the determinant of the matrix is $-14$ .

Importance:
Understanding this basic method of calculating determinants of 2x2 matrices is the first step toward tackling more complex problems involving larger matrices.

Exercise 4.2: Properties of Determinants

Question Example:
Prove that the determinant of the matrix:

B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

is zero.

Solution:
One of the key properties of determinants is that if two rows (or columns) of a matrix are proportional, then the determinant of that matrix is zero. In the matrix $B$ , we notice that:

Row 3 is the sum of Row 1 and Row 2.

R_3 = R_1 + R_2

Since this holds, the determinant of matrix $B$ must be zero.

We can also confirm this by calculating the determinant directly using cofactor expansion. Using the first row for expansion:

|B| = 1 \times \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} - 2 \times \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} + 3 \times \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix}

Each of the 2x2 matrices yields a determinant of zero, so the final determinant is zero.

Exercise 4.3: Cofactors and Minors

Question Example:
Find the minor and cofactor of the element 5 in the following matrix:

C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

Solution:
To find the minor of the element 5, we delete the row and column containing 5 and calculate the determinant of the resulting 2x2 matrix:

\text{Minor of 5} = \begin{vmatrix} 1 & 3 \\ 7 & 9 \end{vmatrix} = (1 \times 9) - (3 \times 7) = 9 - 21 = -12

Next, we find the cofactor of 5, which is the minor multiplied by the appropriate sign factor. The cofactor is given by:

\text{Cofactor of 5} = (-1)^{2+2} \times \text{Minor of 5} = 1 \times (-12) = -12

Thus, the cofactor of 5 is $-12$ .

Exercise 4.4: Applications of Determinants

Question Example:
Using determinants, find the area of a triangle with vertices $A(1, 2)$ , $B(4, 6)$ , and $C(7, 8)$ .

Solution:
The area of a triangle with vertices $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ is given by:

\text{Area} = \frac{1}{2} \times \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|

Substituting the coordinates of the vertices:

\text{Area} = \frac{1}{2} \times \left| 1(6 - 8) + 4(8 - 2) + 7(2 - 6) \right|

= \frac{1}{2} \times \left| 1(-2) + 4(6) + 7(-4) \right|

= \frac{1}{2} \times \left| -2 + 24 - 28 \right| = \frac{1}{2} \times \left| -6 \right| = \frac{1}{2} \times 6 = 3

Thus, the area of the triangle is 3 square units.

Conclusion

Chapter 4 of NCERT Class 12 Mathematics, titled Determinants, provides an in-depth understanding of how to compute determinants and their applications in solving systems of equations, matrix algebra, and geometry. Mastering the concepts and solutions in this chapter is crucial for students aiming to excel in their board examinations. Determinants are not only foundational for understanding more advanced topics in mathematics but also have practical applications in fields like engineering, economics, and computer science.

By thoroughly practicing the exercises provided in this chapter, students can strengthen their problem-solving skills and ensure that they have a solid grasp of the concepts. The NCERT solutions provided here aim to serve as a reliable guide for students to enhance their understanding of determinants and their wide-ranging applications.