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Question #41355

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ matrix with real entries such that all its eigenvalues are real. Let $S = \{x \in \mathbb{R} : \text{tr}(\text{adj}(xI - A)) = 0\}$ . Which of the following statements is \textbf{INCORRECT}?

Options

A.
If $A$ has three distinct eigenvalues, then $S$ has exactly two elements, and for all $x \in S$ , $xI - A$ is non-singular.
B.
If $S$ contains exactly one element $x_0$ , then $x_0I - A$ is a singular matrix.
C.
If $A$ has exactly two distinct eigenvalues, then $S$ has exactly two elements, and for exactly one $x \in S$ , $xI - A$ is singular.
D.
If $S$ contains exactly two elements, then $A$ must have three distinct eigenvalues.

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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

Rolle's Theorem Anchors|Characteristic Equation|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

The question tests the application of Rolle's Theorem to the characteristic polynomial $P(x) = \det(xI - A)$ .

The primary trap is failing to recognize that $\text{tr}(\text{adj}(xI - A))$ is exactly the derivative $P'(x)$ via Jacobi's formula.

Students often incorrectly assume the converse of Rolle's theorem: if the derivative of a cubic has two distinct roots, the cubic itself must have three distinct roots.

This ignores the case of algebraic multiplicity 2 where one of the local extrema touches the x-axis, making the matrix singular at that specific point while still yielding two distinct roots for the derivative.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Rolle's Theorem Anchors
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free