Problem 36 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(A\) be a \(2 \times 2\) matrix Statement-1: \(\operatorname{adj}(\operatorname{adj} A)=A\) Statement-2: \(|\operatorname{adj} A|=|A|\) (A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is true, Statement-2 is true; Statement- 2 is not a correct explanation for Statement-1 (C) Statement-1 is true, Statement- 2 is false (D) Statement-1 is false, Statement- 2 is true

Step-by-Step Solution

Answer

(C) Statement-1 is true, Statement-2 is false.

Step 1: Understand the adjugate and its properties

The adjugate of a matrix, denoted as \(\operatorname{adj}(A)\), is the transpose of the cofactor matrix. For a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the adjugate is \(\operatorname{adj}(A) = \begin{pmatrix} d & -b \ -c & a \end{pmatrix}\). This matrix has the property that \(A \cdot \operatorname{adj}(A) = \det(A) \cdot I\), where \(I\) is the identity matrix.

Step 2: Examine Statement-1 \\( \operatorname{adj}(\operatorname{adj} A) = A \\)

Since \(A\) is a \(2 \times 2\) matrix, the adjugate of an adjugate matrix \(\operatorname{adj}(\operatorname{adj} A)\) equals \(A\) for any such matrix. So, Statement-1 is true for a general \(2 \times 2\) matrix.

Step 3: Investigate Statement-2 \\( |\operatorname{adj} A| = |A| \\)

For a \(2 \times 2\) matrix, the determinant of the adjugate is not equal to the determinant of the original matrix. Instead, \(|\operatorname{adj} A| = (\det A)^{n-1} = (\det A)^1 = |A|\) only when \(\det(A) = 1\) or \(\det(A) = -1\), which is not necessarily true for all \(2 \times 2\) matrices. Thus, Statement-2 is generally false.

Step 4: Determine the correct answer

Since Statement-1 is true for any \(2 \times 2\) matrix and Statement-2 is false as explained, the correct answer is (C) Statement-1 is true, Statement-2 is false.

Key Concepts

Adjugate Matrix

An adjugate matrix, also denoted as \( \operatorname{adj}(A) \), is derived from a given matrix \( A \). To find the adjugate of a matrix, we first need to determine the cofactor of each element of the matrix, then transpose the resulting cofactor matrix. For a \( 2 \times 2 \) matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the adjugate is calculated as follows:

• The cofactor of \( a \) is \( d \), for \( b \) it's \(-c\), for \( c \) it's \(-b\), and for \( d \) it's \( a \).
• Thus, the cofactor matrix is \( \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \).
• Transposing this gives us the adjugate: \( \operatorname{adj}(A) = \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \).

The adjugate has special properties, including that \( A \cdot \operatorname{adj}(A) = \det(A) \cdot I \), where \( I \) is the identity matrix.

Determinant Properties

Determinants are important in understanding various properties of matrices. For a \( 2 \times 2 \) matrix \( A \), the determinant \( \det(A) \) is calculated by the formula \( ad - bc \) where \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \). Some properties of determinants important in this context include:

• The determinant of a product is the product of determinants: \( \det(AB) = \det(A) \cdot \det(B) \).
• If any row or column of a matrix is multiplied by a scalar \( k \), the determinant is also multiplied by \( k \).
• The determinant of an adjugate of \( A \) is \((\det A)^{n-1} \) for an \( n \times n \) matrix. In the case of a \( 2 \times 2 \) matrix, \( (\det A)^1 \).

These properties help us assess matrix behavior in various operations.

2x2 Matrix Determinant

Calculating the determinant of a \( 2 \times 2 \) matrix is a fundamental skill in linear algebra. For matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as:\
\[ \det(A) = ad - bc \] \
This calculation plays a critical role in understanding if matrices are invertible (non-zero determinant means the matrix is invertible). Also, knowing how to find determinants helps with calculating inverses and adjugates, crucial concepts in solving systems of equations and transforming spaces.

Linear Algebra

Linear algebra is a mathematical discipline central to the study of vectors, matrices, and linear transformations. It provides powerful tools for modeling and solving real-world problems using finite-dimensional vector spaces. In particular, matrices such as the \( 2 \times 2 \) matrix discussed are common elements in linear algebra. Key aspects include:

• Matrix operations, which include addition, scalar multiplication, and matrix multiplication.
• The concept of vector spaces and subspaces.
• Applications such as solving systems of linear equations and transformations in various dimensions.

Understanding linear algebra means mastering matrix theory and determinants, enabling the handling of complex computations and transformations in mathematical modeling.

Matrix Theory

Matrix theory is a branch of mathematics focusing on the study of matrices and their properties. A matrix is a rectangular array of numbers, which can represent a myriad of ideas ranging from solving systems of equations to encoding complex transformations in space. In this context, important concepts include:

• Matrix operations: Addition, subtraction, and multiplication.
• Special matrices: Identity, diagonal, symmetric, and adjugate matrices.
• Eigenvalues and eigenvectors as foundational elements for transformation properties.
• Using determinants and inverses to solve linear equations.

The foundational knowledge of matrix theory is essential for applications in physics, computer science, statistics, and many more fields, bridging abstract theory with practical application.