CBSE Class 12 Mathematics Chapter 4 Determinants NCERT Solutions

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NCERT Solutions PDF Class 12 PDF

This chapter provides comprehensive NCERT Solutions for Class 12 Mathematics, focusing on Determinants. Students will learn to evaluate determinants of various orders, understand their properties, and apply them to solve problems. The solutions cover the evaluation of 2x2 and 3x3 determinants, properties of determinants, and their applications in areas like solving systems of linear equations and finding the area of triangles. These solutions are designed to help students grasp the fundamental concepts of determinants, including cofactor expansion and the relationship between determinants and matrices. By working through these detailed explanations and step-by-step problem-solving approaches, students can build a strong foundation in this crucial topic, enhancing their understanding and preparing effectively for their board examinations.

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Quick info

BoardCBSE
ClassClass 12
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 4

Chapter summary

Chapter 4, Determinants, introduces students to the concept of determinants as a scalar value associated with a square matrix. This section provides NCERT Solutions that guide students through calculating determinants for 2x2 and 3x3 matrices. It also covers essential properties of determinants and their applications, such as finding the area of a triangle and solving systems of linear equations. The solutions offer clear, step-by-step methods for evaluating determinants and proving identities related to them.

Learning outcomes

  • Understand the definition and calculation of determinants for 2x2 matrices.
  • Evaluate determinants of 3x3 matrices using cofactor expansion.
  • Apply properties of determinants to simplify calculations and prove identities.
  • Relate determinants to matrices and understand the property |kA| = k^n|A| for a matrix A of order n.
  • Solve problems involving the evaluation of determinants.

Topics covered

Paper topics

  • Determinants of a matrix
  • Evaluation of determinants (2x2)
  • Evaluation of determinants (3x3)
  • Cofactor expansion
  • Properties of determinants
  • Scalar multiplication of matrices and determinants
  • Relationship between |kA| and |A|

Important topics

  • Evaluation of 2x2 determinants
  • Evaluation of 3x3 determinants using cofactor expansion
  • Properties of determinants
  • The property |kA| = k^n|A|

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Questions and Solutions

Question 1

Evaluate the determinant <math>\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}</math>
Solution:

To evaluate the determinant of a 2x2 matrix \begin{vmatrix} a & b \\ c & d \end{vmatrix}, we use the formula $ad - bc$.

For the given determinant, we have $a=2$, $b=4$, $c=-5$, and $d=-1$.

Applying the formula:

\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix} = (2)(-1) - (4)(-5)

= -2 - (-20)

= -2 + 20

= 18

Thus, the value of the determinant is 18.

Question 2

Evaluate the determinants:

(i) \begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix}

(ii) \begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}

Solution:

(i) To evaluate the determinant \begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix}, we apply the formula $ad - bc$.

\begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix} = (\cos \theta)(\cos \theta) - (-\sin \theta)(\sin \theta)

= \cos^2\theta + \sin^2\theta

Using the trigonometric identity $\cos^2\theta + \sin^2\theta = 1$, the value of the determinant is: = 1 (ii) To evaluate the determinant \begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}, we apply the formula $ad - bc$.

\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix} = (x^2 - x + 1)(x + 1) - (x - 1)(x + 1)

We can factor out $(x+1)$ from both terms:

= (x + 1) [(x^2 - x + 1) - (x - 1)]

Simplify the expression inside the brackets:

= (x + 1) [x^2 - x + 1 - x + 1]

= (x + 1) [x^2 - 2x + 2]

Now, expand this expression:

= x(x^2 - 2x + 2) + 1(x^2 - 2x + 2)

= x^3 - 2x^2 + 2x + x^2 - 2x + 2

Combine like terms:

= x^3 + (-2x^2 + x^2) + (2x - 2x) + 2

= x^3 - x^2 + 2

Thus, the value of the determinant is $x^3 - x^2 + 2$.

Question 3

If A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}, then show that |2A| = 4|A|.
Solution:

We are given the matrix A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}. We need to show that |2A| = 4|A|.

First, let's find the matrix $2A$ by multiplying each element of $A$ by 2:

2A = 2 \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix} = \begin{bmatrix} 2 \times 1 & 2 \times 2 \\ 2 \times 4 & 2 \times 2 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 8 & 4 \end{bmatrix}

Now, let's calculate the determinant of $2A$, denoted as $|2A|$:

|2A| = \begin{vmatrix} 2 & 4 \\ 8 & 4 \end{vmatrix}

Using the formula $ad - bc$:

|2A| = (2)(4) - (4)(8)

= 8 - 32

= -24

Next, let's calculate the determinant of $A$, denoted as $|A|$:

|A| = \begin{vmatrix} 1 & 2 \\ 4 & 2 \end{vmatrix}

Using the formula $ad - bc$:

|A| = (1)(2) - (2)(4)

= 2 - 8

= -6

Finally, let's calculate $4|A|$:

4|A| = 4 \times (-6)

= -24

Comparing the values, we see that $|2A| = -24$ and $4|A| = -24$. Therefore, we have shown that $|2A| = 4|A|$. This illustrates the property that for a square matrix $A$ of order $n$, $|kA| = k^n|A|$. In this case, $n=2$, so $|2A| = 2^2|A| = 4|A|$.

Common mistakes

  • Errors in applying the formula for a 2x2 determinant: ad - bc.
  • Sign errors during cofactor expansion for 3x3 determinants.
  • Incorrectly applying scalar multiplication properties to determinants.
  • Calculation mistakes in arithmetic operations within the determinant evaluation.

Revision tips

  • Memorize the formula for a 2x2 determinant and practice its application.
  • Understand the cofactor expansion method thoroughly for 3x3 determinants.
  • Review the properties of determinants, especially those related to row/column operations and scalar multiplication.
  • Work through all solved examples and exercises to reinforce understanding and identify weak areas.

Practice MCQs

Q1. What is the value of the determinant \begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}?

Q2. What is the value of the determinant \begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix}?

Q3. If A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}, what is the value of |A|?

Q4. For a 2x2 matrix A, if we multiply A by a scalar k, how does the determinant |kA| relate to |A|?

Frequently asked questions

What is a determinant in Class 12 Maths?

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix and is used in solving systems of linear equations and other mathematical applications.

How do you calculate the determinant of a 2x2 matrix?

For a matrix \begin{bmatrix} a & b \\ c & d \end{bmatrix}, the determinant is calculated as ad - bc.

What is the key property shown in Question 3 of Exercise 4.1?

Question 3 demonstrates the property |kA| = k^n|A|, where A is a square matrix of order n, and k is a scalar. For a 2x2 matrix, this means |kA| = k^2|A|.

Are the solutions provided for Class 12 Maths Chapter 4 aligned with the CBSE syllabus?

Yes, these solutions are based on the NCERT textbook and are designed to align with the CBSE Class 12 Mathematics syllabus for Chapter 4: Determinants.

How can these NCERT solutions help in exam preparation?

These solutions offer clear, step-by-step explanations for each problem, helping students understand the concepts, practice problem-solving techniques, and identify common mistakes, thereby strengthening their preparation for exams.

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