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Question #41113

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $3 \times 3$ matrices with real entries. A relation $R$ on $S$ is defined by $A R B \iff \det(A) = \det(B)$ . Let $[M]$ denote the equivalence class of a matrix $M \in S$ . Consider the matrix $P = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix}$ . If a matrix $A \in [P]$ satisfies the equation $A \cdot \text{adj}(A) + 3A = A^2 + k I$ for some real number $k$ , which of the following statements is ALWAYS true?

Options

A.
$k = 0$ and the maximum possible trace of $A$ is $6$
B.
$k = 0$ and the maximum possible trace of $A$ is $9$
C.
$k = 27$ and the maximum possible trace of $A$ is $9$
D.
$k = -27$ and the maximum possible trace of $A$ is $6$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Equivalence Class Partitioning|Inversion and Determinant Product|Inverse Existence Criterion

Why This Question Is Tricky

Common Mistakes

This question tests the conceptual boundary of equivalence classes and matrix properties.

The trap lies in the 'Inverse Existence Criterion'.

Students often ignore the equivalence class constraint ( $A \in [P] \implies \det(A) = 0$ ) and assume $A$ can be any matrix satisfying the equation.

If they assume $A$ is invertible (e.g., $A = 3I$ ), they will incorrectly calculate $k = 27$ and a maximum trace of $9$ (Option C).

Even if they correctly find $k=0$ , forgetting that the singular nature of $A$ prevents the eigenvalues from being $\{3,3,3\}$ leads to the trap of maximum trace $9$ (Option B).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Equivalence Class Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free