Problem 32 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \neq Q\). 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: Analyze Given Conditions

We are given that \(P^3 = Q^3\) and \(P^2 Q = Q^2 P\). These equations imply certain relations between the matrices \(P\) and \(Q\) which might suggest they are related in symmetry or commutativity.

Step 2: Assess the Given Problem

The exercise asks for the determinant of \(P^2 + Q^2\). From the properties of determinants and polynomials, if two matrices commute, it often leads to simplifications when considering their powers or sums.

Step 3: Use Matrix Identities and Commutativity

Given that \(P^2 Q = Q^2 P\), assume that \(P\) and \(Q\) commute to investigate the properties. Let's assume \(P = A\) and \(Q = -A\) for some matrix \(A\). Then \(P^3 - Q^3 = 0\) simplifies as both yield zero: \((P = A)\rightarrow A^3 - (-A)^3 = 0\).

Step 4: Compute the Determinant

With \(P = A\) and \(Q = -A\), \(P^2 + Q^2 = A^2 + (-A)^2 = 0\). Hence, the determinant of \(P^2 + Q^2\) is:\[det(P^2 + Q^2) = det(0) = 0\]Predictably, when P and Q are negations of one another, this results.

Step 5: Analyze Other Possibilities

As our assumption leads to a coherent match fulfilling all conditions and the determinant equating to zero without inconsistency, we verify this against the given options.

Key Concepts

Matrix Commutativity

Matrix commutativity is a concept that deals with the ability of matrices to be multiplied in any order with the same result. For two matrices \(A\) and \(B\), commutativity means that \(AB = BA\). This property is not true for all pairs of matrices, unlike numbers where multiplication is always commutative.
In our exercise, it's given that \(P^2 Q = Q^2 P\), which suggests a level of commutativity between matrices \(P\) and \(Q\). This specific relationship, however, doesn’t imply full commutativity (i.e., \(PQ = QP\)) but rather a commutation under certain conditions such as specific polynomial forms.

• Commutative matrices usually lead to simplifications when calculating powers or sums of matrices.
• The conditions given in the problem hint towards matrices \(P\) and \(Q\) sharing some properties leading to symmetry in their products.

Such properties can often be exploited to simplify calculations, like finding determinants or solving matrix equations.

Matrix Equations

Matrix equations are crucial in understanding relationships between different matrices. They involve operations such as addition, multiplication, and determination of inverse matrices. Here, we are given two matrix equations: \(P^3 = Q^3\) and \(P^2 Q = Q^2 P\).
Matrix equations often involve solving for unknown matrices or establishing a specific identity. The provided equations suggest certain properties or assumptions, as seen in this exercise. For instance, the equality \(P^3 = Q^3\) implies that, given the appropriate context, matrices \(P\) and \(Q\) might be negations or scalar multiples of one another when simplified in polynomial forms.
The second equation, \(P^2 Q = Q^2 P\), shows an interesting dependence on matrix powers. This is not a standard commutation but rather an interaction that appears to lead to the exploration of deeper relationships between \(P\) and \(Q\).

• Such equations are gateways to understanding matrix properties, like rank, nullity, and even eigenvalues.
• In solving these equations, sometimes assumptions like commutativity or specific matrix transformations help simplify the problem.

Properties of Determinants

The determinant is a special number that can be calculated from a square matrix. It provides important properties about the matrix, such as whether the matrix is invertible. A determinant of zero suggests that the matrix is singular, meaning it does not have an inverse.
In this exercise, we've been asked to compute the determinant of \(P^2 + Q^2\). Using the assumptions made from prior steps, particularly that \(P = A\) and \(Q = -A\), it simplifies to \(det(P^2 + Q^2) = det(0)\). This implies that the matrix \(P^2 + Q^2\) is singular.

• The additive property of determinants doesn’t exist, so \(det(A + B) eq det(A) + det(B)\).
• However, when matrices are fully commutative, certain patterns in solving for determinants can be recognized.

Exploring determinant properties allows us to reveal vital characteristics of matrices, including those relevant for solving systems of linear equations or understanding matrix transformations.