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Question #40382

Difficulty: 8/10 | 🟢 Free

The Question

Let $B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ . Let $S$ be the set of all real numbers $x$ for which there exists an invertible matrix $A$ of order 2 with real entries such that $A B - B^2 A = ([x]^2 - 5[x] + 6)I$ , where $I$ is the identity matrix of order 2 and $[ \cdot ]$ denotes the greatest integer function. Find the set $S$ .

Options

A.
[2, 4)
B.
\emptyset
C.
[2, 3)
D.
$(-\infty, 2) \cup [4, \infty)$

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

The Empty Set Trap|Inverse Existence Criterion|Greatest Integer Function (GIF)|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

The problem tests the student's understanding of matrix non-commutativity and the existence of inverses.

A common trap is assuming $AB = BA$ .

If a student makes this assumption, the equation simplifies to $(I-B)BA = kI$ .

Taking determinants gives $\det(I-B)\det(B)\det(A) = k^2$ .

Since $\det(I-B) = 0$ , this forces $k = 0$ , leading to $(I-B)A = 0$ , which implies $\det(A) = 0$ .

This contradicts the requirement that $A$ is invertible, leading the student to incorrectly conclude that no such $x$ exists (the 'Empty Set' trap).

By correctly treating $A$ and $B$ as non-commutative, a valid family of invertible matrices is found, yielding a non-empty solution set for $x$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	The Empty Set Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free