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Question #40893

Difficulty: 8/10 | 🟢 Free

The Question

Let $A, B, C$ be three points in $\mathbb{R}^3$ represented as column vectors. Let $M$ be the $3 \times 3$ matrix whose columns are $A, B, C$ respectively. It is given that $M$ is a symmetric orthogonal matrix with $\det(M) = 1$ and $M \neq I_3$ . The sum of all elements of $M$ is equal to $3$ . Points $P, Q, R$ are obtained by dividing the directed line segments $AB, BC, CA$ externally in the ratio $2:1$ respectively. Let $N$ be the $3 \times 3$ matrix whose columns are $P, Q, R$ respectively. Find the trace of the matrix $N$ .

Options

A.
1
B.
3
C.
5
D.
-1

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

Section Formula Internally/Externally|Elementary Row Operation Invariance|Matrix Trace Properties

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between affine coordinate transformations and linear matrix invariants.

The primary trap lies in assuming that because the points $P, Q, R$ are formed via an affine section formula (where the sum of coefficients is $2-1=1$ ), the trace of the new matrix $N$ will trivially equal the trace of $M$ .

Students often conflate the invariance of trace under similarity transformations ( $P M P^{-1}$ ) with right-multiplication coordinate shifts ( $M K$ ).

By forcing the evaluation of $Tr(M K)$ , the problem requires students to realize that the absolute trace properties are highly sensitive to cyclic coordinate displacements, demanding the extraction of off-diagonal sums using the properties of symmetric orthogonal matrices.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Section Formula Internally/Externally
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free