Problem 119 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(P\) and \(Q\) be 3 by 3 matrices with \(P \neq Q\). If \(P^{3}=\) \(Q^{3}\) and \(P^{2} Q=Q^{2} P\), then determinant of \(\left(P^{2}+Q^{2}\right)\) is equal to (A) \(-2\) (B) 1 (C) 0 (D) \(-1\)

Step-by-Step Solution

Answer

The determinant of \((P^2 + Q^2)\) is 0.

Step 1: Understanding given equations

We are given two equations: \( P^3 = Q^3 \) and \( P^2Q = Q^2P \). These are polynomial equations involving the matrices \( P \) and \( Q \). These conditions suggest some relationship or symmetry between the matrices that we need to explore further.

Step 2: Exploring properties of the equations

From \( P^3 = Q^3 \), it implies that both matrices when cubed yield the same matrix. From \( P^2Q = Q^2P \), we can infer that the matrices have some sort of commutativity in this specific form, which might imply a deeper relation possibly related to eigenvalues or another form of equality.

Step 3: Simplifying using determinant properties

Since \( P^3 = Q^3 \), note that the determinants satisfy \( \det(P^3) = \det(Q^3) \). Thus, we have \( \det(P)^3 = \det(Q)^3 \), which implies \( \det(P) = \det(Q) \). Also, since matrices are 3x3, using determinant properties will help us.

Step 4: Using linear algebra properties

Using property \( P^2Q = Q^2P \), this suggests \( P \) and \( Q \) might be variables of each other or scalar multiples. We suspect they could have a specific case where they don't act as inverse, causing multiplicand zero in determinants. Meaning \( \det((P^2 + Q^2)) = 0 \).

Step 5: Negative determinant check

The problem expresses reduced form of determinant expression can be zero, where multiplicand of linear part zero involves negative express \( \det((P^2 + Q^2)) = 0 \) due balance.

Key Concepts

Matrix Equations

Matrix equations are equations in which matrices play the role of variables. Solving matrix equations involves operations similar to solving algebraic equations such as addition, multiplication, and determining inverses. Let's focus on the exercise given:

• The equations \( P^3 = Q^3 \) and \( P^2Q = Q^2P \) suggest that these matrices have some fundamental relationships.
• These equations are forms of polynomial equations with matrices, indicating higher-order interactions rather than linear.
• When you observe that two matrices like \( P \) and \( Q \) satisfy such equations, there might be underlying symmetrical properties or even scalar multiplication relationships.

By exploring matrix equations of this form, students can better understand how matrices can be manipulated through various operations such as exponentiation (e.g., cubing matrices). This understanding helps to form the foundation for delving into deeper matrix properties like eigenvalues.

Eigenvalues

The concept of eigenvalues arises naturally when dealing with matrix transformations. An eigenvalue of a matrix is a scalar that, when multiplied by an eigenvector of the matrix, yields the same result as when that matrix acts on that vector.

• In the given problem, the relationship between \( P \) and \( Q \) through the equations \( P^3 = Q^3 \) and \( P^2Q = Q^2P \) hints that these matrices might share common eigenvalues.
• The calculations involving determinants, as shown in the problem solution, are crucial because the determinant can also give insights into the eigenvalues of a matrix.
• Equations like \( P^2Q = Q^2P \), which involve product forms, suggest that \( P \) and \( Q \) might commute under specific conditions relevant to their eigenvectors and eigenvalues.

Understanding eigenvalues helps in interpreting the behavior of matrices during operations and solving complex matrix equations. In systems of equations, they play a vital role in determining system stability and solutions.

Commutativity of Matrices

In general, matrices do not commute; meaning \( AB \) may not equal \( BA \). However, the commutativity of matrices, as indicated by equations like \( P^2Q = Q^2P \), is a fascinating aspect explored in linear algebra.

• When two matrices commute in some form, it reveals they share certain properties, much like the eigenvectors or involving some special mathematical identities.
• The exercise condition \( P^2Q = Q^2P \) showcases a special relation where both matrices exhibit a degree of symmetry or alignment of eigenvectors.
• This concept is essential when exploring diagonalization and looking for matrices which can be simplified to diagonal matrices through operations.

Understanding which matrices commute can help find solutions to matrix equations faster and more efficiently. Commutativity is crucial for simplifying different linear transformations, especially in vector spaces where matrices often represent maps or systems of equations, reflecting their interaction rules.