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Question #41339

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x)$ be a $3 \times 3$ matrix with differentiable entries such that $A(1) = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{pmatrix} \quad \text{and} \quad A'(1) = \begin{pmatrix} 1 & 2 & -1 \\ 0 & 1 & 3 \\ 2 & 0 & 1 \end{pmatrix}$ Let $f(x) = \det(A(x))$ . If $g(x)$ is the local inverse function of $f(x)$ in a neighborhood of $x=1$ , what is the value of $g'(-8)$ ?

Options

A.
$-\frac{1}{12}$
B.
$\frac{1}{12}$
C.
$-\frac{1}{8}$
D.
$\frac{1}{8}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Inverse Function Derivative|Differentiation of Determinants|Singular vs Non-Singular Matrices|System of Linear Equations (Cramer/Inverse)

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of the inverse function derivative rule and Jacobi's formula for determinant differentiation.

The primary trap lies in attempting to expand the determinant symbolically, which is impossible since A(x) is not explicitly given, or misapplying the chain rule for matrices (e.g., incorrectly assuming f'(x) = det(A'(x))).

Students must recognize that d/dx det(A) = tr(adj(A) dA/dx), which is equivalent to the Frobenius inner product of the cofactor matrix and the derivative matrix.

Furthermore, the problem implicitly tests the non-singularity condition; because f'(1) is non-zero, the local inverse is guaranteed to exist and be differentiable.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Inverse Function Derivative
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free