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Question #41098

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be a $5 \times 5$ upper triangular matrix with entries $p_{ij} = \binom{j-1}{i-1} q^{i-1} (1-q)^{j-i}$ for $1 \le i \le j \le 5$ , where $q \in (0,1)$ . Let $M$ be a $5 \times 5$ matrix with entries $m_{ij} = i^2 + j^2 + (i-j)^5$ for $1 \le i, j \le 5$ . If $M = S + K$ is the unique decomposition of $M$ into a symmetric matrix $S$ and a skew-symmetric matrix $K$ , and $B = P^{-1} K P$ , find the value of: $\det(B^3 + B) + \text{Tr}(P^{-1} S P) + \log_q(\det(P))$

Options

A.
110
B.
120
C.
130
D.
140

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

Symmetry and Invariance|Symmetric and Skew-Symmetric Decomposition|Triangular/Diagonal Matrix Shortcut

Why This Question Is Tricky

Common Mistakes

This question tests the invariance of matrix properties under similarity transformations and the spectral properties of skew-symmetric matrices.

The primary trap is the exhaustive computation of the decomposed matrices $S$ and $K$ , followed by the basis transformation $P^{-1}KP$ .

Students often fail to recognize that $B^3+B$ is similar to $K^3+K$ , and since $K$ is a skew-symmetric matrix of odd order ( $5 \times 5$ ), any odd polynomial of $K$ without a constant term remains skew-symmetric.

Consequently, its determinant is strictly zero.

Furthermore, the problem exploits the cyclic property of the trace ( $\text{Tr}(P^{-1}SP) = \text{Tr}(S)$ ) and the triangular matrix shortcut for determinants, bypassing the need to invert the complex probability matrix $P$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Symmetry and Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free