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Question #72214

Difficulty: 8/10 | 🟢 Free

The Question

Let $\mathcal{M}_3(\mathbb{R})$ be the vector space of all $3 \times 3$ real matrices. Let $A \in \mathcal{M}_3(\mathbb{R})$ be a matrix whose characteristic polynomial is $P(\lambda) = (\lambda - 2)^3$ and whose minimal polynomial is $M(\lambda) = (\lambda - 2)^2$ . Consider the system of differential equations given by the operator equation $dX = (AX - XA)dt$ , where $X(t) \in \mathcal{M}_3(\mathbb{R})$ . Let $\mathcal{S}$ be the subspace of solutions of the form $X(t) = C e^{kt}$ , where $C$ is a constant non-zero matrix and $k \in \mathbb{R}$ . What is the dimension of $\mathcal{S}$ ?

Options

A.
3
B.
5
C.
6
D.
9

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

Exact Differential Recognition|Linear Independence vs Dependence|Characteristic Equation

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

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

Difficulty Boost : +4

Common Mistakes

This question tests the student's ability to analyze a matrix differential equation by treating the commutator $L(X) = AX - XA$ as a linear operator.

The primary trap is assuming the solution space dimension is either 3 (confusing matrix ODEs with vector ODEs) or 9 (assuming the operator is diagonalizable since its only eigenvalue is 0).

Another trap is incorrectly calculating the dimension of the commutant by assuming a single Jordan block, which would yield a dimension of 3.

By specifying the minimal polynomial as $(\lambda - 2)^2$ , the problem forces students to construct the correct Jordan canonical form and apply the Rank-Nullity theorem to find the exact geometric multiplicity of the operator, which is 5.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Exact Differential Recognition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free