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Question #41856

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} e^x & x \\ 0 & e^x \end{pmatrix}$ and $B(x) = \begin{pmatrix} e^{-x} & 0 \\ x & e^{-x} \end{pmatrix}$ be two matrix-valued functions. Define $U(x) = A(x)B(x)$ and $V(x) = B(x)A(x)$ . If the Wronskian matrix of $U$ and $V$ is defined as $W(x) = U(x)V'(x) - U'(x)V(x)$ , find the value of $\det(W(2))$ .

Options

A.
$-16$
B.
$0$
C.
$16$
D.
$-112$

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

Algebra of Functions|Linear Independence vs Dependence|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the non-commutativity of matrix multiplication within the context of differential operators, specifically the Wronskian.

A common trap for students is to assume that algebraic products of functions behave commutatively ( $AB = BA$ ), which would lead to $U(x) = V(x)$ and incorrectly evaluating the Wronskian as $U U' - U' U$ , yielding a determinant of $-112$ .

An even more severe conceptual error is assuming matrices commute with their own derivatives, which would result in a Wronskian of $0$ .

By forcing the explicit calculation of $U V' - U' V$ , the problem rigorously evaluates the student's ability to handle order-dependent product rules in matrix calculus.

Quick Facts About This Question

Question Information

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