Determinants

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Class 12 Maths Chapter 4
Determinants
Important Questions

Preparing for the Class 12 Maths exams, particularly Chapter 4 on Determinants, holds immense significance in establishing a robust mathematical foundation. This chapter delves into the realm of determinants, elucidating their properties, operations, and applications within mathematical problem-solving. To excel in this chapter, familiarizing oneself with essential questions becomes paramount. Here are some class 12 Determinants important questions and answers. These questions are meticulously crafted to assess the grasp of fundamental concepts. Embracing a diverse range of question styles empowers students to navigate uncertainties, ensuring thorough preparedness for the impending exams. Engaging with these questions not only bolsters confidence but also refines the art of problem-solving, an indispensable skill in mastering the complexities of mathematics.”

Table of Contents

Introduction

The concept of determinants is a fundamental aspect of linear algebra and is widely used in mathematics, physics, engineering, and computer science. A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, including whether it is invertible and properties related to its linear transformations. Class 12 Determinants important questions and answers will help students understand the chapter in a better way.

What are Determinants?

Determinants are a mathematical concept used primarily in the field of linear algebra. They are associated with square matrices and have a wide range of applications in various areas of mathematics, physics, and engineering. Determinants are defined only for square matrices (matrices with the same number of rows and columns). For larger square matrices, the determinant is calculated using more complex methods, such as expansion by minors or cofactors. determinant of a matrix is a single numerical value.

Class 12 Determinants Important Questions and Answers

Q1. If A is a square matrix of order 3 and |A| = 5, then the value of |2A′| is

Options

(a) -10
(b) 10
(c) -40
(d) 40

Ans. (d) 40
Explanation:
According to the property of transpose of a matrix, (kA′) = kA′
Also, from the property of determinant of a matrix, |A′| = |A|
and $|kA| = k^n|A|$ , where n is the order of matrix A.
Thus, $|2A'| = 2^3|A|$ {since A is a square matrix of order 3} = 8 × 5 = 40

Q2. Which of the following is correct?

Options

(a) Determinant is a square matrix.
(b) Determinant is a number associated with a matrix.
(c) Determinant is a number associated with a square matrix.
(d) None of these

Ans. (c) Determinant is a number associated with a square matrix.

Q3. Given that A is a square matrix of order 3 and |A| = -4, then |adj A| is equal to

Explanation:
Given that A is a square matrix of order 3 and |A| = -4.
We know that $|adj A| = |A|^{n−1},$ where n is the order of matrix A.
So, $|adj A| = (−4)^{3-1} = (-4)^2 = 16$

Q4. Given that $\text{A = } [a_{ij}]$ is a square matrix of order 3×3 and |A| = -7, then the value of

$\displaystyle{\sum_{i = 1}^3} \text{ a}_{i2} \text{A}_{i2} \text{ where A}_{ij}$ denotes the cofactor of element
$a_{ij}$

Explanation:
|A| = -7
Order of matrix A is 3×3.
Now, $\displaystyle{\sum_{i = 1}^3} a_{i2}\space A_{i2} = \space a_{12}\space A_{12} + a_{22}\space A_{22}\space + \space a_{32}\space A_{32}\space \\[4.5 bp]$ = |A|
= -7

Q5. If A is an invertible matrix of order 2, then det $(A^{-1})$ is equal to

Explanation:
Given that the A is an invertible matrix of order 2.
If the matrix is invertible, then its determinant is not equal to 0.
We know that,
$AA^{-1} = I$ , where I is the identity matrix
Taking determinant on both sides,
$\\[4.5 bp]|AA^{-1}| = |I|\\[4.4 bp] |A| |A^{-1}| = 1\\[4.5 bp] |A^{-1}| = \dfrac{1}{|A|}$ {since A is non-singular, |A| ≠ 0}

CBSE Class 12 Maths Chapter wise Important Questions

Chapter No.	Chapter Name
Chapter 1	Relations and Functions
Chapter 2	Inverse Trigonometric Functions
Chapter 3	Matrices
Chapter 4	Determinants
Chapter 5	Continuity and Differentiability
Chapter 6	Applications of Derivatives
Chapter 7	Integrals
Chapter 8	Applications of the Integrals
Chapter 9	Differential Equations
Chapter 10	Vectors
Chapter 11	Three - dimensional Geometry
Chapter 12	Linear Programming
Chapter 13	Probability

Conclusion

Determinants are a fundamental concept in mathematics, particularly in the field of linear algebra. They provide a scalar value associated with square matrices and have a wide range of applications across various fields. Exploring cbse class 12 maths Determinant important questions, becomes essential for a comprehensive grasp of this fundamental aspect of mathematics. To enhance understanding, platforms like Oswal.io provide a wealth of resources, including question-answer sets, comprehensive class 12 questions and answers.Utilising these resources ensures students acquire the knowledge necessary to excel in their studies and examinations.

Frequently Asked Questions

Q1 : What is a Determinant?

Ans: A determinant is a scalar value that is calculated from a square matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor it represents in geometric transformations.

Q2 : What Does a Zero Determinant Indicate?

Ans: A zero determinant indicates that the matrix is singular, meaning it does not have an inverse. Geometrically, it implies that the transformation represented by the matrix collapses the space into a lower dimension.

Q3 : How Does the Determinant Relate to Eigenvalues?

Ans: Determinants are used in the calculation of eigenvalues. Specifically, the eigenvalues of a matrix are the roots of its characteristic polynomial, which is obtained by taking the determinant of the matrix minus a scalar multiple of the identity matrix.

Q4 : Are Determinants Used in Solving Systems of Linear Equations?

Ans: Yes, determinants are used in Cramer’s Rule to solve systems of linear equations. This rule provides a way to find the solution of a system using the determinants of matrices.

Q5: Can Determinants Be Used for Non-Square Matrices?

Ans: No, determinants are defined only for square matrices. For non-square matrices, other concepts such as rank or singular value decomposition are used.

Chapter Wise Important Questions for CBSE Board Class 12 Maths
Relations and Functions
Inverse Trigonometric Functions
Matrices
Determinants
Continuity and Differentiability
Applications of Derivatives
Integrals
Applications of the Integrals
Differential Equations
Vectors
Three - dimensional Geometry
Linear Programming
Probability

Class 12 Maths Chapter 4 Determinants Important Questions

Introduction

What are Determinants?

Class 12 Determinants Important Questions and Answers

Q1. If A is a square matrix of order 3 and |A| = 5, then the value of |2A′| is

Options

(a) -10 (b) 10 (c) -40 (d) 40

Q2. Which of the following is correct?

Options

(a) Determinant is a square matrix. (b) Determinant is a number associated with a matrix. (c) Determinant is a number associated with a square matrix. (d) None of these

Q3. Given that A is a square matrix of order 3 and |A| = -4, then |adj A| is equal to

Q4. Given that \text{A = } [a_{ij}] is a square matrix of order 3×3 and |A| = -7, then the value of

\displaystyle{\sum_{i = 1}^3} \text{ a}_{i2} \text{A}_{i2} \text{ where A}_{ij} denotes the cofactor of element a_{ij}

Q5. If A is an invertible matrix of order 2, then det (A^{-1}) is equal to

CBSE Class 12 Maths Chapter wise Important Questions

Conclusion

Frequently Asked Questions

Q1 : What is a Determinant?

Q2 : What Does a Zero Determinant Indicate?

Q3 : How Does the Determinant Relate to Eigenvalues?

Q4 : Are Determinants Used in Solving Systems of Linear Equations?

Q5: Can Determinants Be Used for Non-Square Matrices?

Class 12 Maths Chapter 4
Determinants
Important Questions

(a) -10
(b) 10
(c) -40
(d) 40

(a) Determinant is a square matrix.
(b) Determinant is a number associated with a matrix.
(c) Determinant is a number associated with a square matrix.
(d) None of these

Q4. Given that $\text{A = } [a_{ij}]$ is a square matrix of order 3×3 and |A| = -7, then the value of

$\displaystyle{\sum_{i = 1}^3} \text{ a}_{i2} \text{A}_{i2} \text{ where A}_{ij}$ denotes the cofactor of element
$a_{ij}$

Q5. If A is an invertible matrix of order 2, then det $(A^{-1})$ is equal to