Before You Begin

Question #41282

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix where its elements are given by the integral: $a_{ij} = \int_{0}^{\pi} \frac{(i^2 - j^2) \sin x}{1 + e^{x - \pi/2}} dx$ Find the value of $\det(I - A^2)$ , where $I$ is the $3 \times 3$ identity matrix.

Options

A.
9801
B.
9604
C.
10000
D.
9409

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

Skew-Symmetric Determinant|Determinant Product Rule|Even and Odd Function Symmetry|Substitution (Change of Variable)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize definite integral properties with advanced matrix spectral theory.

The primary trap is twofold: First, the integral appears intimidating due to the shifted exponential, masking the simple evaluation via the $x \to \pi - x$ property.

Second, even after successfully evaluating $a_{ij} = i^2 - j^2$ , students often attempt to compute $A^2$ and its determinant by brute-force matrix multiplication.

This is a massive time-sink and highly prone to arithmetic errors.

The intended 'hidden structure' path requires recognizing $A$ as a skew-symmetric matrix and leveraging its eigenvalues ( $0, \pm i\lambda$ ) to instantly compute the determinant of the polynomial matrix $I - A^2$ using the spectral mapping theorem.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Skew-Symmetric Determinant
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free