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Question #73584

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{c} = 3\vec{a} - \vec{b}$ and $\vec{a} \times \vec{b} = 2\hat{i} - \hat{j} + 2\hat{k}$ . Let $A$ be a $3 \times 3$ matrix whose columns are the vectors $\vec{a} \times \vec{b}$ , $\vec{b} \times \vec{c}$ , and $\vec{c} \times \vec{a}$ respectively. Let $\alpha$ be the largest real eigenvalue of $A$ . If $B = A^3 - 6A^2 + 2I$ , where $I$ is the $3 \times 3$ identity matrix, find the value of $\det(B - \alpha I)$ .

Options

A.
1100
B.
-1100
C.
1225
D.
850

Check Your Answer

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

Geometric Interpretation of Cross Product|Characteristic Equation|Singular vs Non-Singular Matrices

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 integration of vector cross product properties with matrix algebra and eigenvalue theory.

The primary trap lies in the construction of the matrix A.

Students who fail to recognize that the vectors are coplanar will attempt to assign arbitrary variables to a, b, and c, leading to an unsolvable mess.

Even if they construct A, calculating the characteristic polynomial via standard determinant expansion is tedious and prone to error.

The conceptual shortcut is recognizing that a matrix with proportional columns has a rank of 1, meaning its non-zero eigenvalue is simply its trace.

Furthermore, the problem tests the application of the Cayley-Hamilton theorem to reduce high-degree matrix polynomials, and the property that the determinant of a matrix is the product of its eigenvalues, bypassing the need to explicitly compute the matrix A^2 - 5I.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Geometric Interpretation of Cross Product
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free