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Question #41208

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 3 & -6 \\ 1 & -2 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 2 \\ -2 & 6 \end{pmatrix}$ . Let $Y(t)$ be a $2 \times 2$ matrix-valued function satisfying the differential equation $\frac{dY}{dt} = AY - YA$ with the initial condition $Y(0) = B$ . Find the sum of the elements of the matrix $Y(\ln 2)$ .

Options

A.
37
B.
10
C.
2
D.
-33

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Idempotent and Involutory Properties|Linear Differential Equation Structure|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question tests the resolution of matrix differential equations and the application of idempotent matrix properties to matrix exponentials.

The core trap is the 'Integration Constant Trap' combined with non-commutativity.

Students often treat matrix ODEs like scalar ODEs, incorrectly assuming $\frac{dY}{dt} = AY - YA \implies \frac{dY}{dt} = 0$ (yielding a sum of 10), or they integrate to $Y(t) = e^{At}B$ by ignoring the right-sided multiplication (yielding 2).

A correct approach requires recognizing the commutator structure to find $Y(t) = e^{At} B e^{-At}$ , and then leveraging $A^2=A$ to evaluate the exponentials exactly without infinite series, punishing those who fail to maintain strict matrix ordering.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Idempotent and Involutory Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free