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Question #41831

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ be a $2 \times 2$ matrix where the entries $a, b, c, d$ are chosen independently and uniformly at random from the set $\{1, 2, 3, 4, 5, 6\}$ . Given that the trace of matrix $A$ is $7$ , what is the probability that the rows of $A$ are linearly dependent?

Options

A.
$\frac{5}{54}$
B.
$\frac{5}{108}$
C.
$\frac{43}{648}$
D.
$\frac{1}{18}$

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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

Combinatorial Probability|Linear Dependency of Rows|Matrix Trace Properties

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of combinatorial probability and linear algebra.

The specific trap lies in the conditional probability space created by the matrix trace property.

Students often calculate the unconstrained probability of linear dependency ( $\det(A)=0$ ) or incorrectly assume the trace constraint $\text{Tr}(A)=7$ applies symmetrically to all elements, failing to realize it strictly partitions the sample space into $216$ valid matrices instead of $1296$ .

By forcing the trace to a specific integer, the degrees of freedom for the diagonal elements are reduced, requiring a systematic enumeration of the determinant condition $ad=bc$ strictly within the constrained subsets of $ad \in \{6, 10, 12\}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Combinatorial Probability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free