Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let A be any \(3 \times 3\) invertible matrix. Then which one of the following is not always true? \(\quad\) [Online April 8, 2017] (a) \(\operatorname{adj}(\mathrm{A})=|\mathrm{A}| \cdot \mathrm{A}^{-1}\) (b) \(\operatorname{adj}(\operatorname{adj}(\mathrm{A}))=|\mathrm{A}| \cdot \mathrm{A}\) (c) \(a d j(a d j(A))=|A|^{2} \cdot(a d j(A))^{-1}\) (d) \(\operatorname{adj}(\operatorname{adj}(\mathrm{A}))=|\mathrm{A}| \cdot(\operatorname{adj}(\mathrm{A}))^{-1}\)

Step-by-Step Solution

Answer

Option (d) is not always true.

Step 1: Understanding the Adjugate of Matrix A

The adjugate (or adjoint) of a matrix \( A \), denoted as \( \operatorname{adj}(A) \), is the transpose of its cofactor matrix. For a \( 3 \times 3 \) matrix, the relationship between \( A \), its determinant \( |A| \), and its adjugate is given by \( A \cdot \operatorname{adj}(A) = |A|I \), where \( I \) is the identity matrix.

Step 2: Verifying Option (a)

Option (a) states that \( \operatorname{adj}(A) = |A| \, A^{-1} \). Using the relationship \( A \cdot \operatorname{adj}(A) = |A|I \) and the definition of the inverse \( A^{-1} = \frac{1}{|A|} \operatorname{adj}(A) \), we find that this statement is always true for invertible matrices.

Step 3: Verifying Option (b)

Option (b) suggests that \( \operatorname{adj}(\operatorname{adj}(A)) = |A| \cdot A \). By properties of adjugates, \( \operatorname{adj}(\operatorname{adj}(A)) = |A|^{n-2} A \) for an \( n \times n \) matrix. For a \( 3 \times 3 \) matrix, this simplifies to \( |A|A \) when \( n = 3 \). Thus, option (b) is true.

Step 4: Verifying Option (c)

Option (c) claims \( \operatorname{adj}(\operatorname{adj}(A)) = |A|^2 \cdot (\operatorname{adj}(A))^{-1} \). We know from earlier steps that \( \operatorname{adj}(\operatorname{adj}(A)) = |A|A \). Since \( (\operatorname{adj}(A))^{-1} = \frac{1}{|A|} A \), the right-hand side becomes \( |A| A \), which agrees with our earlier findings. Hence, option (c) is true.

Step 5: Verifying Option (d)

Option (d) indicates \( \operatorname{adj}(\operatorname{adj}(A)) = |A| \cdot (\operatorname{adj}(A))^{-1} \). From earlier, \( \operatorname{adj}(\operatorname{adj}(A)) = |A|A \) and \((\operatorname{adj}(A))^{-1} = \frac{1}{|A|} A \), thus \(|A| \cdot (\operatorname{adj}(A))^{-1} = A \). Therefore, this does not hold because the two sides do not equate. Thus, option (d) is not always true.

Key Concepts

Adjugate Matrix

An adjugate matrix, also known as the adjoint of a matrix, is an essential tool in matrix algebra. It's formed by taking the cofactor matrix of a given square matrix and then transposing it. Think of it as rearranging the components of a matrix in a special way that links it closely to the original matrix's determinants.

For a given matrix \( A \), its adjugate is denoted as \( \operatorname{adj}(A) \). A key identity associated with the adjugate for any invertible matrix is:

• \( A \cdot \operatorname{adj}(A) = |A|I \)

Here, \( |A| \) is the determinant of \( A \) and \( I \) is the identity matrix. This formula reveals that when you multiply the original matrix by its adjugate, you end up with a scaled version of the identity matrix, where the scaling factor is the determinant.

The adjugate becomes particularly useful when calculating a matrix's inverse, playing a pivotal role in the formula for inversion.

Determinant

The determinant is a scalar value that provides essential insights into certain properties of a matrix. It can be seen as a special ultra-condensed summary of the matrix, useful in determining whether a matrix is invertible or not. For a matrix \( A \), the determinant is represented as \( |A| \).

When the determinant is zero, it means the matrix is singular, or non-invertible. In other terms, there is no inverse matrix, and certain matrix operations like solving linear equations can become undefined.

For a \( 3 \times 3 \) matrix, for instance, calculating the determinant involves a formula connecting its elements which typically involves sums and products of its minors and cofactors.

• The formula involves creating a cofactor matrix and operating with these cofactors and the elements from the first row or column.

Understanding determinants helps in exploring deeper concepts in linear algebra, such as eigenvalues and eigenvectors, and can be beneficial when interpreting transformations represented by matrices.

Inverse Matrix

An inverse matrix is akin to the opposite of a matrix; capable of undoing the transformation that the original matrix applies. If matrix \( A \) is invertible, its inverse is denoted \( A^{-1} \), and they satisfy the equation:

• \( A \cdot A^{-1} = I \)
• \( A^{-1} \cdot A = I \)

The identity matrix \( I \) is the equivalent of 1 in matrix multiplication. The existence of \( A^{-1} \) depends on \( A \) being non-singular, i.e., its determinant \( |A| eq 0 \).

A practical formula for finding the inverse, especially in 3x3 matrices, relies on the adjugate matrix and the determinant:

• \( A^{-1} = \frac{1}{|A|} \operatorname{adj}(A) \)

This equation uses the determinant to "scale" the adjugate, reflecting the requirement of the determinant to be non-zero. Understanding and using inverse matrices is vital in solving systems of linear equations, where it provides a direct method to find solutions by utilizing matrix algebra.