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Question #67692

Difficulty: 8/10 | 🟢 Free

The Question

Consider the following two statements for differentiable $n \times n$ matrix functions $A(t)$ and $B(t)$ defined for $t > 0$ :

Statement I: If the matrices satisfy the implicit relation $\text{Tr}\left( \frac{d}{dt}(t A(t)) \right) - 2 \text{Tr}(A(t)) = t^2 \text{Tr}(B(t))$ , then the explicit trace is given by $\text{Tr}(A(t)) = t \int \text{Tr}(B(t)) dt + C t$ , where $C$ is an arbitrary constant.

Statement II: The trace operator satisfies $\text{Tr}\left( \frac{d}{dt}(t A(t)) \right) = t \frac{d}{dt}\text{Tr}(A(t))$ , and the integrating factor for the resulting linear differential equation in $\text{Tr}(A(t))$ is $t^{-2}$ .

Which of the following logically follows?

Options

A.
\text{Statement I is True, Statement II is True; Statement II is a correct explanation for Statement I}
B.
\text{Statement I is True, Statement II is True; Statement II is NOT a correct explanation for Statement I}
C.
\text{Statement I is True, Statement II is False}
D.
\text{Statement I is False, Statement II is True}

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

Matrix Trace Properties|Linear Differential Equation Structure|Implicit to Explicit Conversion

Why This Question Is Tricky

Examiner Levers

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the causal relationship between matrix calculus properties and linear differential equations.

The primary trap lies in the implicit-to-explicit conversion of the trace operator.

Students often incorrectly assume that the trace of a derivative of a scalar-multiplied matrix, $\text{Tr}(\frac{d}{dt}(tA))$ , is simply $t \frac{d}{dt}\text{Tr}(A)$ , failing to apply the product rule within the trace operator.

This error leads to a missing $\text{Tr}(A)$ term, which alters the coefficient in the resulting linear differential equation from $-1/t$ to $-2/t$ .

Consequently, the student computes an incorrect integrating factor of $t^{-2}$ instead of $t^{-1}$ , compounding the error and leading them to falsely believe Statement II is true and Statement I is false.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Matrix Trace Properties
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free