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Question #40501

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ matrix with positive integer eigenvalues. Let $M_1 = \begin{pmatrix} \text{tr}(A) & -1 \\ |A| & 0 \end{pmatrix}$ and $M_2 = \begin{pmatrix} \text{tr}(\text{adj}(A)) & -1 \\ |\text{adj}(A)| & 0 \end{pmatrix}$ . If $M_1$ and $M_2$ share a common eigenvalue and $|A| \neq 1$ , find the value of $\text{tr}(A) + |A| + \text{tr}(\text{adj}(A))$ .

Options

A.
15
B.
17
C.
19
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

Condition for Common Roots|Characteristic Equation|Adjoint-Determinant Relationship

Why This Question Is Tricky

Common Mistakes

The problem tests the synthesis of matrix properties and the theory of equations.

The trap lies in the Adjoint-Determinant relationship: students often correctly identify $|\text{adj}(A)| = |A|^2$ but fail to relate $\text{tr}(\text{adj}(A))$ to the elementary symmetric polynomials of the eigenvalues.

A common mistake is assuming the matrices $M_1$ and $M_2$ must be identical for them to share an eigenvalue, leading to $e_1 = e_2$ and $e_3 = e_3^2$ , which incorrectly forces $|A|=1$ (violating the given condition).

The correct path requires deriving the resultant condition for the two quadratics and solving the resulting Diophantine equation for the eigenvalues.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Condition for Common Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free