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Question #41303

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ non-zero matrix with real entries such that $A^2 = \mathbf{0}$ . Let $M(\lambda) = (\lambda - 2)A + (\lambda^2 - 4)A^2$ . A new matrix $B(\lambda)$ is formed by first exchanging the first and third rows of $M(\lambda)$ , and then exchanging the first and second columns of the resulting matrix. If $\lambda \in \mathbb{R}$ satisfies the equation $\det(B(\lambda)) + (\lambda - 2)^2 (\lambda - 1) = (\lambda - 2)^2 (\lambda^2 - 9\lambda + 20)$ then the sum of all possible values of $\lambda$ is:

Options

A.
10
B.
12
C.
14
D.
21

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 Zero-Factor Cancellation Trap|Row/Column Exchange Parity|Nilpotent Matrix Logic

Why This Question Is Tricky

Common Mistakes

This question targets the 'Zero-Factor Cancellation Trap' intertwined with nilpotent matrix properties and determinant transformations.

Students often treat determinant equations purely algebraically and hastily cancel common factors like $(\lambda - 2)^2$ from both sides.

By doing so, they implicitly assume the factor is non-zero, losing the valid root $\lambda = 2$ .

Furthermore, the problem tests the understanding that a nilpotent matrix of order 2 has a determinant of zero, and that two parity-reversing operations (row and column swaps) leave the determinant's sign unchanged.

The trap is sprung when students cancel the factor before evaluating the identically zero determinant, leading them to the incorrect sum of 10 instead of 12.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	The Zero-Factor Cancellation Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free