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Question #40358

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = [a_{ij}]$ be a square matrix of order $235$ , where $a_{ij} = |i-j|^{234} \cos(i-j) + |i-j|^{235} \sin(i-j)$ for all $1 \le i, j \le 235$ . Let $A = P + Q$ , where $P$ is a symmetric matrix and $Q$ is a skew-symmetric matrix. What is the value of $\det(Q^{2024} + Q^{2025}) + \text{Tr}(Q^{2025})$ ?

Options

A.
0
B.
1
C.
235
D.
2025

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Modulus Function Symmetry|Symmetric and Skew-Symmetric Decomposition|Determinant Product Rule

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to identify symmetric and skew-symmetric components hidden behind modulus and trigonometric functions.

The core trap lies in the 'Determinant Product Rule'.

Students often attempt to expand or find patterns in the matrix elements, missing the fundamental properties of skew-symmetric matrices.

By asking for $\det(Q^{2024} + Q^{2025})$ , the problem forces the student to factor the expression into $\det(Q^{2024})\det(I+Q)$ to utilize the property that $\det(Q)=0$ for odd-order skew-symmetric matrices.

Additionally, asking for the trace of an odd power tests the invariant property that odd powers of skew-symmetric matrices remain skew-symmetric, thus having a zero trace.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Modulus Function Symmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free