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Question #40934

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ real skew-symmetric matrix such that $\text{Tr}(A A^T) = 28$ . Let $M(x) = I + \sin(x) A + \tan^2(x) A^T A$ , where $I$ is the $3 \times 3$ identity matrix. Evaluate the limit: $L = \lim_{x \to 0} \left( \det(M(x)) \right)^{\frac{1}{x^2}}$

Options

A.
$e^{42}$
B.
$e^{28}$
C.
$e^{14}$
D.
Does not exist

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

The 1^∞ Form Shortcut|Properties of Transpose Determinant|Matrix Equality

Why This Question Is Tricky

Common Mistakes

This question tests the $1^\infty$ limit form combined with matrix properties, specifically targeting the 'Matrix Equality' trap.

Students often confuse the limit of a scalar determinant with the determinant of a matrix limit.

If a student attempts to evaluate the matrix limit $\lim_{x \to 0} M(x)^{\frac{1}{x^2}}$ using the matrix exponential $e^{\lim \frac{M(x)-I}{x^2}}$ , they will encounter the term $\frac{\sin x}{x^2} A$ , which diverges.

This leads them to incorrectly conclude that the limit 'Does not exist'.

However, the determinant is a scalar function; the divergent $O(1/x)$ terms in the matrix eigenvalues are complex conjugates that cancel out upon multiplication, resulting in a perfectly well-defined scalar limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	The 1^∞ Form Shortcut
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free