Problem 72 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let A be a \(2 \times 2\) matrix Statement \(-1: \operatorname{adj}(\operatorname{adj} \mathrm{A})=\mathrm{A}\) Statement \(-\mathbf{2}: \mid\) adj \(\mathrm{A}|=| \mathrm{A} \mid \quad\) [2009] (a) Statement- 1 is true, Statement- 2 is true. Statement-2 is not a correct explanation for Statement- 1 . (b) Statement- 1 is true, Statement- 2 is false. (c) Statement \(-1\) is false, Statement- 2 is true. (d) Statement- 1 is true, Statement \(-2\) is true. Statement- 2 is a correct explanation for Statement-1.

Step-by-Step Solution

Answer

Statement-1 is true, Statement-2 is false, so option (b) is correct.

Step 1: Understand the Problem

The exercise presents two statements involving properties of matrices, and we need to determine the truth value of each statement and the relationship between them.

Step 2: Review Key Concepts

Recall that the adjugate of a matrix \( A \), denoted as \( \text{adj}(A) \), is a matrix of co-factors, and the relation \( \text{adj}(\text{adj}(A)) = A \) holds for 2x2 matrices. Additionally, the property \( |\text{adj}(A)| = |A| \) generally does not hold for a 2x2 matrix.

Step 3: Evaluate Statement 1

For a \(2 \times 2\) matrix \( A \), it is known that \( \text{adj}(\text{adj}(A)) = A \). Thus, Statement 1 is true because it is a known property for 2x2 matrices.

Step 4: Evaluate Statement 2

Statement 2 involves the determinant of the adjugate and the original matrix. For a \(2 \times 2\) matrix, \(|\text{adj}(A)| = |A|\) is not a true statement because the determinant of the adjugate scales differently. Therefore, Statement 2 is false.

Step 5: Compare Statement Relations

Since Statement 1 involves a property of the double adjugate and Statement 2 involves determinant properties, and Statement 2 is incorrect, it cannot serve as an explanation for Statement 1.

Step 6: Choose the Correct Option

Given the analysis, Statement 1 is true, Statement 2 is false. Thus the correct option is (b): Statement-1 is true, Statement-2 is false.

Key Concepts

Adjugate of a Matrix

The adjugate of a matrix, often referred to as the adjoint, is essential in advanced matrix operations. For a given matrix \( A \), the adjugate is denoted as \( \operatorname{adj}(A) \). It is constructed by taking the transpose of the cofactor matrix of \( A \). For a \(2 \times 2\) matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the adjugate, \( \operatorname{adj}(A) \), is calculated by swapping the diagonal elements and changing the signs of the off-diagonal elements:

• Original: \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)
• Adjugate: \( \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \)

Importantly, for any \(2 \times 2\) matrix, the property \( \operatorname{adj}(\operatorname{adj}(A)) = A \) holds true. This means that if you compute the adjugate twice, you get back to the original matrix. Understanding and applying the adjugate is crucial in solving systems of linear equations using the inverse matrix method, particularly because it helps in calculating the inverse of a non-singular matrix.

Determinant Properties

The determinant function is a pivotal concept in linear algebra. For a matrix \( A \), the determinant is represented as \( |A| \) and provides critical insights into the matrix's properties.For a \(2 \times 2\) matrix, \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is given by the formula:\[|A| = ad - bc\] This scalar value can offer information about the matrix, such as whether it is invertible. If \(|A| = 0\), the matrix is singular, which means it does not have an inverse.Regarding statement ": \(|\operatorname{adj}(A)| eq |A|\) for a \(2 \times 2\) matrix": the determinant of an adjugate does not equal the determinant of the original matrix in general. Instead, there is a unique relationship where \(|\operatorname{adj}(A)| = |A|^{n-1}\), where \( n \) is the size of the matrix. Thus, for a \(2 \times 2\) matrix, it fundamentally highlights the discrepancy between the determinants, guiding us to conclude that they are different unless specified otherwise by specific matrix properties.

Matrix Theory

Matrix theory forms the backbone of linear algebra and is crucial in various mathematical and applied fields, such as computer science, physics, and engineering. It involves the study of matrices, their properties, operations that can be performed on them, and their applications. An important aspect of matrix theory is understanding various matrix operations like addition, multiplication, finding the inverse, and more complex concepts like calculating the adjugate and determinant. These operations can help solve systems of linear equations, perform transformations, and much more. Moreover, understanding different matrix types—such as square, diagonal, and identity matrices—and their properties is fundamental in efficient computation and application. For example, a square matrix (where the number of rows and columns is equal) is especially pertinent in operations like finding the determinant or inverse.

• Consistency in notation and operation is critical.
• Grasping the geometric interpretation of a matrix can offer deeper insights, such as viewing matrix multiplication as transformation in space.

Matrix theory, therefore, not only involves pure theoretical exploration but also practical application to solve real-world problems.