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Question #66838

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 2k & 3-k & 4 \end{pmatrix}$ for some real parameter $k$ . Let $B = A^2 - 4A + (k-4)I$ , where $I$ is the $3 \times 3$ identity matrix. If $S$ is the set of all real values of $k$ for which $B$ is a singular matrix, find the product of all elements in $S$ .

Options

A.
-1
B.
0
C.
1
D.
$-\frac{7}{4}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Characteristic Equation|Inversion and Determinant Product|Denominator Restriction

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the 'Denominator Restriction' trap within the context of the Cayley-Hamilton theorem.

Students frequently manipulate the characteristic equation to express a polynomial in $A$ in terms of $A^{-1}$ (e.g., $B = 2k A^{-1} - I$ ).

They then set $\det(B) = 0$ to find $k$ , which yields a cubic equation with a root at $k=0$ .

The fatal flaw is failing to recognize that the algebraic step dividing by $2k$ (or multiplying by $A^{-1}$ ) implicitly assumes $\det(A) \neq 0$ , which requires $k \neq 0$ .

Including the extraneous root $k=0$ leads to the trap answer of 0.

Rigorous case analysis is required to verify the boundary condition where the matrix becomes singular.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Characteristic Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free