Determinants – Complete Guide for Class 12 Math Chapter 4

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The chapter “Determinants” in Class 12th Math delves into the concept of determinants, a fundamental tool for solving systems of linear equations and analyzing the properties of matrices. It introduces the definition and calculation of determinants for 2×2 and 3×3 matrices, properties such as the effect of row operations on determinants, and the concept of the minor, cofactor, and adjugate matrices. The chapter also covers the application of determinants in finding the inverse of matrices, solving linear equations using Cramer’s rule, and determining the consistency of systems of equations. Understanding determinants is crucial for advanced mathematical topics, including linear algebra and calculus, and has practical applications in areas such as physics, engineering, and computer science.

Determinants

Introduction

Determinants are a fundamental concept in linear algebra that associates every square matrix with a numerical value or expression. This value, known as the determinant, provides critical insights into the properties of the matrix, including whether it is invertible.

Understanding Determinants

A determinant is denoted by det(A) or |A| for a square matrix A of order n. For a matrix A with elements aij, the determinant is constructed by applying specific rules based on the order of the matrix.

Determinant of a Matrix of Order One

If A is a square matrix of order 1, such as:

            A = [ a₁₁]

The determinant of A is simply the value of a_{11}:

det(A) = a₁₁ 

Determinant of a Matrix of Order Two

For a matrix A of order 2 x 2:

                      A =  

 The determinant is given by:

           det(A) = a₁₁a₂₂ – a₁₂a₁₂

 Determinant of a Matrix of Order Three

For a 3 x 3 matrix:

    A        =  

​

The determinant can be expanded as:

det(A) = a₁₁ ( a₂₂a₃₃ – a₂₃a₃₂) – a₁₂ ( a₂₁a₃₃ – a₂₃a₃₁) + a₁₃ ( a₂₁a₃₂ – a₂₂a₃₁)

Properties of Determinants

Determinants have several essential properties that simplify calculations and provide deeper insights into matrix operations.

• Unchanged Value on Interchanging Rows and Columns:

The determinant remains the same if the rows and columns of the matrix are interchanged.

• Sign Change on Interchanging Two Rows or Columns:

If any two rows (or columns) of a determinant are interchanged, the sign of the determinant changes.

• Zero Value for Identical Rows or Columns:

If any two rows (or columns) of a determinant are identical, the determinant equals zero.

            Example:

                              Δ   =  =   0

This happens because interchanging identical rows does not change the determinant, leading to a zero value.

• Multiplying a Row or Column by a Constant:

If each element of a row (or a column) of a determinant is multiplied by a constant k, the determinant itself is multiplied by k.

• Addition of Elements as a Sum:

If the elements of a row or column are expressed as the sum of two or more terms, the determinant can be written as the sum of two or more determinants.

• Equimultiples of Corresponding Elements: 

Adding equimultiples of corresponding elements from another row or column does not change the determinant’s value.

           Example

                          Δ      = = 0

Here, applying column operations shows that the determinant is zero due to identical rows or columns.

Applications of Determinants

Determinants are used in various applications, such as finding the collinearity of three points, solving systems of linear equations, and evaluating the invertibility of matrices.

• Minor of an Element

The minor of an element aij in a determinant is found by deleting the ith row and the j-th column containing aij. The resulting determinant is called the minor of aij, denoted by Mij.

• Cofactor of an Element

The cofactor of an element aij, denoted by Aij, is calculated as:

Aᵢⱼ = (−1)ᶤ⁺ʲ × Mᵢⱼ

Adjoint and Inverse of a Matrix

• Adjoint of a Matrix: Let be a square matrix of order n and let Aij be a cofactor of aij in A. Then the transpose of the matrix of the cofactors of the elements of A is called the adjoint of A. It is denoted by adj A.

                                                    A      =  

and adj A = 

        Aᵀ      =  ​

Theorem: If A is any given square matrix of order n, then 

A (adjA) = (adjA) A= |A|I.

Definition: A square matrix A is said to be singular if |A| = 0. 

Definition: A square matrix A is considered non-singular if |A| ≠ 0.

1. Non-Singular Matrices (Theorem): If A and B are non-singular matrices of the same order, then both AB and BA are also non-singular matrices of the same order.

2. Determinant of Product(Theorem): For square matrices A and B of the same order, the determinant of their product is the product of their determinants: det⁡(AB)=det⁡(A)⋅det⁡(B)

Note   If A is a non-singular matrix of order n, then 

                        |adj A| = |A|ⁿ⁻¹

• The inverse of a Matrix: A square matrix A is invertible if it is non-singular, meaning det(A) ≠ 0. The inverse of A is given by:

               A⁻¹ = 1/det(A) ×adj(A)

Examples:

Example 1: For matrices 

A      = 

and    B    =    

verify that (AB)⁻¹ = B⁻¹ A⁻¹ 

Find A⁻¹ :  A⁻¹ = 1/ det(A) x adj(A) = 

    1/1 A⁻¹ = 

Find B⁻¹ :  B⁻¹ = 1/ det(B) x adj(B) = 1/-1

  B⁻¹ =      

Compute AB: AB = 

x

= 

Find (AB)⁻¹:  (AB)⁻¹ = 1/det(AB) x adj(AB)

Verify:  (AB)⁻¹ = B⁻¹ A⁻¹

x  

= 

Applications of Determinants and Matrices

Key Points:

• Consistent system: A system of equations is said to be consistent if its solution (one or more) exists. 
• Inconsistent system: A system of equations is said to be inconsistent if its solution does not exist.

Solution of Systems of Linear Equations

To solve a system of linear equations using matrices, represent the system as 

AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the constants matrix.

For example, solve the system: 

5x−7y= 27x−5y = 3

This can be written in matrix form as: 

x​

          = 

1. Find det(A): det(A) = 5⋅(−5)−(−7⋅7) = −25 + 49 = 24

2. Compute the inverse A⁻¹: A⁻¹ = 1/24 

3. Find X using X = A⁻¹ B:X = 1/24 

                                                      x 

= 

Thus, the solution to the system is x = 11/24 and  y = 1/24 

Let’s Conclude

In summary, Chapter 4 – Determinants of CBSE Class 12 Mathematics is a vital component of your mathematical toolkit. It equips you with the necessary skills to understand the properties and applications of determinants, enabling you to solve complex systems of linear equations and analyze matrices effectively. As you dive deeper into determinants, remember that grasping these concepts will enhance your proficiency in advanced topics such as linear algebra and calculus, which have far-reaching applications in fields like physics, engineering, and computer science.

By utilizing the resources provided by iPrep, including animated videos, practice questions, and detailed notes, you can master Chapter 4 – Determinants with confidence. Embrace this learning journey and leverage these tools to ensure your success not only in examinations but also in applying mathematical concepts in real-world scenarios. Keep exploring, practicing, and reinforcing your understanding of determinants to excel in your academic pursuits!