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Question #41015

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $3 \times 3$ diagonal matrices whose principal diagonal elements are chosen from the set $\{0, 1\}$ . Two matrices $A$ and $B$ are chosen randomly and independently from $S$ with replacement. Let $E$ be the event that $\det(2A^T B) = 0$ and $F$ be the event that $\det(A - B) = 0$ . The probability $P(E \cap F)$ is equal to

Options

A.
$\frac{55}{64}$
B.
$\frac{441}{512}$
C.
$\frac{7}{8}$
D.
$\frac{63}{64}$

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

Mutually Exclusive vs Independent|Transpose of Product|Determinant Scaling Property

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between independent events and events with overlapping probability spaces derived from matrix properties.

The primary trap lies in the assumption of independence.

A student might calculate $P(E) = 63/64$ and $P(F) = 7/8$ , and incorrectly assume that the determinant of a product and the determinant of a difference are independent events, leading to the trap answer $441/512$ .

The lethal collision occurs when analyzing the boundary cases of matrix singularity: the complements of these events ( $E^c$ and $F^c$ ) are actually mutually exclusive.

Recognizing that $E^c \cap F^c = \emptyset$ drastically alters the probability space, forcing $P(E \cup F) = 1$ and revealing the true intersection probability.

It also subtly tests the determinant scaling property $\det(kA) = k^n\det(A)$ to ensure students don't get derailed by the scalar multiplier.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Mutually Exclusive vs Independent
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free