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Question #66474

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ matrix with real entries such that $AA^T = I$ , $A^2 = I$ , and $\text{Tr}(A) = 1$ . Let $\vec{r}_1, \vec{r}_2, \vec{r}_3$ be the vectors formed by the first, second, and third rows of $A$ respectively. Let $\vec{u}$ be a vector along the internal angle bisector of $\vec{r}_1$ and $\vec{r}_2$ with magnitude $2$ . Let $\vec{v}$ be a vector along the internal angle bisector of $\vec{r}_2$ and $\vec{r}_3$ with magnitude $\sqrt{2}$ . If $\vec{w} = \vec{u} \times \vec{v}$ , then the value of $\sqrt{2} \vec{w} \cdot (3\vec{r}_1 - 4\vec{r}_2 + 5\vec{r}_3)$ is:

Options

A.
$-24$
B.
$24$
C.
$-8$
D.
$8$

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

Orthogonal Matrix Conditions|Angle Bisector Direction|Collinearity vs Coplanarity

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 properties of orthogonal matrices and their geometric interpretation as 3D coordinate frames.

The primary trap lies in the assumption that all orthogonal matrices represent rotations (where $\det(A) = 1$ and rows form a right-handed system).

By providing the conditions $A^2 = I$ and $\text{Tr}(A) = 1$ , the student is forced to deduce that the eigenvalues are $1, 1, -1$ , meaning $\det(A) = -1$ .

This indicates an improper rotation (reflection), which means the row vectors form a left-handed coordinate system.

Consequently, all cross products of the basis vectors have inverted signs (e.g., $\vec{r}_1 \times \vec{r}_2 = -\vec{r}_3$ ).

Failing to recognize this left-handed orientation leads directly to the trap answer of 24.

Quick Facts About This Question

Question Information

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