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Question #40345

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 1 & -1 & 2 \\ 2 & 1 & -1 \\ 4 & -1 & 3 \end{pmatrix}$ and $M(x) = A + x I$ for $x \in \mathbb{R}$ , where $I$ is the $3 \times 3$ identity matrix. Let $P(x)$ be a matrix such that $M(x) P(x) = I$ for all $x > 0$ . Evaluate the limit: $\lim_{x \to 0^+} \frac{1}{x} \det(I - A P(x))$

Options

A.
$\frac{1}{5}$
B.
0
C.
5
D.
\text{Does not exist}

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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

Identity Function Behavior|Inversion and Determinant Product|Singular vs. Non-Singular Threshold|Adjoint-Determinant Relationship

Why This Question Is Tricky

Common Mistakes

This question tests the behavior of the inverse of a matrix as it approaches a singular state (Trap 230).

Students often incorrectly assume that since $M(x)P(x) = I$ , the limit of $P(x)$ behaves continuously or acts like $A^{-1}$ , leading them to erroneously conclude that $I - AP(x) \to 0$ and the determinant is $0$ .

The trap exposes the failure to recognize that $P(x)$ is undefined at $x=0$ (since $A$ is singular) and requires algebraic manipulation using the identity $I - AP(x) = xP(x)$ before applying determinant properties.

Furthermore, it tests the precise calculation of the characteristic polynomial coefficients, specifically relying on the trace of the adjoint matrix being exactly zero to yield a finite, non-zero limit.

Quick Facts About This Question

Question Information

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