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Question #41243

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x)$ be a $3 \times 3$ orthogonal matrix with continuously differentiable entries for all $x \in \mathbb{R}$ . Let $M(x)$ be a $3 \times 3$ matrix with differentiable entries satisfying the differential equation $M'(x) = \left( \int_{x}^{x^2} A(t)^T A'(t) dt \right) M(x)$ for all $x \in \mathbb{R}$ , with $M(1) = A(1)$ . Find the value of $\frac{d}{dx} \left( \det(M(x)) \right)$ at $x = 2024$ .

Options

A.
0
B.
1
C.
2024
D.
-1

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Orthogonal Matrix Conditions|Differentiation of Determinants|Leibniz Rule for Differentiation under Integral

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of matrix differential equations, properties of orthogonal matrices, and Jacobi's formula for the derivative of a determinant.

The primary trap lies in the integral $\int_{x}^{x^2} A(t)^T A'(t) dt$ .

Students who attempt to explicitly find $M(x)$ or differentiate the integral using the Leibniz rule will face a massive computational hurdle, as the limits are functions of $x$ and the integrand involves matrix derivatives.

The elegant solution requires recognizing that $A(t)^T A'(t)$ is skew-symmetric due to the orthogonality of $A(t)$ , meaning its trace is zero.

By applying Jacobi's formula, $\frac{d}{dx} \det(M) = \det(M) \text{tr}(M' M^{-1})$ , the integral of the trace evaluates to zero, immediately yielding a derivative of zero without any complex integration.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Orthogonal Matrix Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free