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Question #41258

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{pmatrix}$ and let $f(x) = \cos(\sin^{-1} x) + \ln(1 + \tan^{-1} x)$ . Evaluate $\text{Tr}(A^T f(A))$ , where $\text{Tr}(M)$ denotes the trace of matrix $M$ and $A^T$ is the transpose of $A$ .

Options

A.
0
B.
3
C.
87
D.
90

Check Your Answer

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

Composition Inverse Identity Traps|Matrix Trace Properties|Nilpotent Matrix Logic

Why This Question Is Tricky

Common Mistakes

This question triggers the 'Identity Trap' by presenting a matrix with elements $>1$ inside inverse trigonometric and logarithmic functions.

Students often incorrectly assume the matrix Maclaurin series diverges, forgetting that convergence depends strictly on the spectral radius (eigenvalues).

Since $A$ is nilpotent, its eigenvalues are 0, guaranteeing convergence and terminating the infinite series exactly at $A^2$ .

Furthermore, the problem tests trace properties via the Frobenius inner product $\text{Tr}(A^T B) = \sum A_{ij}B_{ij}$ .

Students frequently misapply trace linearity or erroneously assume $\text{Tr}(f(A)) = f(\text{Tr}(A))$ , leading to distractors like 0 or 90.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Composition Inverse Identity Traps
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free