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Question #41795

Difficulty: 8/10 | 🟢 Free

The Question

Let $V = \{ \vec{u}, \vec{v}, \vec{w} \}$ be a set of three vectors where $\vec{u} = [1, -2, 1]$ , $\vec{v} = [-1, 1, 2]$ , and $\vec{w} = [2, -1, -3]$ . Let $S$ be the set of all $3 \times 3$ matrices whose rows are formed by choosing vectors from $V$ with replacement. A matrix $A$ is chosen at random from $S$ . Let $E_1$ be the event that the system of linear equations $A \vec{x} = \vec{0}$ has non-trivial solutions, and $E_2$ be the event that $A$ is a nilpotent matrix. The value of $P(E_2 \mid E_1)$ is:

Options

A.
$\frac{1}{7}$
B.
$\frac{1}{9}$
C.
$\frac{2}{21}$
D.
$\frac{1}{3}$

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

Sample Space Partitioning|Nilpotent Matrix Logic|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bypass brute-force matrix calculations by applying theoretical properties of determinants and nilpotent matrices.

The primary trap is attempting to evaluate the determinant for all 27 possible matrices.

Students must recognize that since the base vectors are linearly independent, a zero determinant strictly requires repeated rows, reducing the sample space via basic combinatorics.

Furthermore, students often forget that nilpotency strictly requires the trace to be zero.

By using the trace as a primary filter, the search space for nilpotent matrices collapses from 21 to just 3, which can then be quickly verified.

A common mistake is assuming $\det(A)=0$ and $\text{Tr}(A)=0$ is sufficient for $3 \times 3$ nilpotency without verifying the principal minors, though here the setter has carefully chosen the vectors such that the trace-zero matrices are indeed nilpotent, rewarding the logical leap.

Quick Facts About This Question

Question Information

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