Before You Begin

Question #41929

Difficulty: 8/10 | 🟢 Free

The Question

Let $u$ and $v$ be two unit vectors in $\mathbb{R}^3$ such that the angle between them is $\frac{\pi}{6}$ . Let $P = uu^T$ and $Q = vv^T$ be the orthogonal projection matrices onto the one-dimensional subspaces spanned by $u$ and $v$ , respectively. Define the transformation matrix $T = PQ - QP$ . Which of the following represents the minimal polynomial $m(\lambda)$ of $T$ ?

Options

A.
$\lambda^3 + \frac{3}{16}\lambda$
B.
$\lambda^3 - \frac{3}{16}\lambda$
C.
$\lambda^3 + \frac{3}{8}\lambda$
D.
$\lambda^3 - \frac{3}{8}\lambda$

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

Dot Product Projection Sign|Linear Independence vs Dependence|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of projection matrices in a non-commutative space and the properties of skew-symmetric transformations.

The primary trap lies in the misapplication of the transpose property for matrix products.

A student might incorrectly assume $(PQ)^T = P^T Q^T$ , leading to $T^T = P^T Q^T - Q^T P^T = PQ - QP = T$ .

This false assumption of symmetry would imply that $T$ has real eigenvalues, tricking the student into selecting the polynomial with a negative linear coefficient (Option B).

By correctly applying $(PQ)^T = Q^T P^T$ , one reveals that $T$ is skew-symmetric ( $T^T = -T$ ), which mandates purely imaginary eigenvalues and a minimal polynomial with a positive linear coefficient.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Dot Product Projection Sign
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free