Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If adj \(B=A\) and \(P, Q\) are two unimodular matrices, i.e., \(|P|=1=|Q|\), then \(\left(Q^{-1} B P^{-1}\right)^{-1}\) is equal to (A) \(P A Q\) (B) \(P B Q\) (C) \(Q A P\) (D) \(Q B P\)

Step-by-Step Solution

Answer

(A) \(P A Q\)

Step 1: Understand the Given Information

We are given that \( \text{adj} \, B = A \) and \( P \text{ and } Q \) are unimodular matrices, meaning \( |P| = 1 \) and \( |Q| = 1 \). We have to find the expression for \( \left(Q^{-1} B P^{-1}\right)^{-1} \).

Step 2: Work with Matrix Inverses

Recall that the inverse of a product of matrices can be expressed by reversing their order and taking the inverse of each. Therefore, \( \left(Q^{-1} B P^{-1}\right)^{-1} = P B^{-1} Q \).

Step 3: Substitute \( \text{adj} \, B = A \)

Since \( \text{adj} \, B = A \), we know that \( B^{-1} = \frac{1}{\det(B)}A \), given that \( B \cdot \text{adj} \, B = \det(B)I \). However, \( B \) should be invertible, implying \( B^{-1} \cdot B = I \). Thus, we conclude \( B^{-1} = A \), since \( \det(B)=1 \) for \( \text{adj}\, B = A \).

Step 4: Substituting into the Matrix Expression

Substitute \( B^{-1} = A \) (considering \( \det(B)=1 \)) and substitute back into the expression: \( \left(Q^{-1} B P^{-1}\right)^{-1} = P A Q\).

Step 5: Conclusion

The simplified expression \( P A Q \) corresponds to option (A) in the given answer choices.

Key Concepts

Unimodular Matrix

A unimodular matrix is a fascinating concept in the world of matrix algebra. Essentially, a matrix is considered unimodular if its determinant is either +1 or -1.

• Property of Inverses: Unimodular matrices have a unique feature: they are invertible, and their inverses are also unimodular matrices.
• Determinant: The determinant of a unimodular matrix remains constant under multiplication with another unimodular matrix, specifically being either +1 or -1.
• Effect on Volume: In a geometrical sense, the transformation represented by a unimodular matrix does not alter the volume of the space it acts upon.

Unimodular matrices are frequently used in computations involving integer matrices, especially in lattice-based algorithms, where preserving volume is crucial. In the given exercise, both matrices \(P\) and \(Q\) are unimodular, which plays a key role in manipulation and inversion tasks in the problem.

Adjugate Matrix

The adjugate matrix, often symbolized as \( \text{adj} \, B \), plays an important role in finding inverses of matrices. It is essentially the transpose of the cofactor matrix of \(B\).

• Role in Inversion: If \( \det(B) eq 0\), the inverse of a matrix \(B\) can be calculated using its adjugate with the formula: \( B^{-1} = \frac{1}{\det(B)} \text{adj} \, B \).
• Connection to Determinant: The adjugate matrix provides a pivotal relationship between a matrix and its determinant, being a core component in expressing the inverse.
• Exercise Example: Given \( \text{adj} \, B = A \), we derive that \( B^{-1} = A \) because for \( \text{adj} \, B = A \), the assumption here is \( \det(B) = 1 \).

Understanding the adjugate fills the gap between a matrix and its inverse, enabling deep insights into matrix equations and simplifying complex operations, as illustrated in this problem.

Matrix Inverse

Finding a matrix inverse is a foundational operation in linear algebra. A matrix \(B\) is invertible if there exists another matrix, \(B^{-1}\), such that the product \(B B^{-1} = I\), where \(I\) is the identity matrix.

• Calculation: As previously mentioned, if \( \det(B) eq 0\), the inverse of \(B\) can be neatly found as \( B^{-1} = \frac{1}{\det(B)} \text{adj} \, B \).
• Properties: Matrix inverses exhibit properties like \( (AB)^{-1} = B^{-1}A^{-1} \), crucial for simplifying expressions involving multiple matrices.
• Exercise Insight: The exercise leverages the property of inverses to transform \( \left(Q^{-1} B P^{-1}\right)^{-1} \) into \( P A Q \), utilizing the traits of the given matrices and known relationships, simplifying it further to match the option (A).

Mastering the concept of matrix inverses not only solves equations efficiently but also has wide applications in computer graphics, engineering, and other scientific fields.