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Question #40615

Difficulty: 8/10 | 🟢 Free

The Question

Consider a directed graph with 3 vertices $V_1, V_2, V_3$ . Let $A = [a_{ij}]_{3 \times 3}$ be a matrix where $a_{ij}$ represents the number of directed paths from $V_i$ to $V_j$ . It is given that $a_{ij} = \binom{j}{i}$ for $i \le j$ , and $a_{ij} = i-j$ for $i > j$ . If $B = \text{adj}(A)$ , find the value of $\text{Tr}(AB) - \text{Tr}(BA^T)$ .

Options

A.
0
B.
6
C.
15
D.
24

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

Geometry-based Combinations|Properties of Transpose Determinant|Adjoint-Determinant Relationship

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to construct a matrix from combinatorial rules and apply properties of the adjoint and trace.

The primary trap is the assumption that an adjacency matrix representing paths is symmetric ( $A = A^T$ ), a common heuristic from undirected graph theory.

If a student falls for this, they will incorrectly assume $BA^T = BA = |A|I$ , leading to $\text{Tr}(BA^T) = 15$ and a final answer of 0.

It also rewards the efficient computation of $\text{Tr}(XY^T)$ as the sum of element-wise products (Hadamard product sum), bypassing tedious full matrix multiplication.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Geometry-based Combinations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free