Before You Begin

Question #40911

Difficulty: 8/10 | 🟢 Free

The Question

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation representing a reflection across the plane $x + y + z = 0$ , followed by a uniform scaling by a factor of $4$ . Let $A$ be the standard matrix representation of $T$ , and $C$ be the cofactor matrix of $A$ . If $\vec{v} = 2\hat{i} - \hat{j} + 3\hat{k}$ , what is the sum of the scalar projections of the vector $\vec{w} = C\vec{v}$ onto the coordinate axes?

Options

A.
64
B.
-64
C.
16
D.
-16

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

Coordinate Axis Projections|Scalar Multiplication Geometric Scaling|Minors and Cofactors Sign Convention

Why This Question Is Tricky

Common Mistakes

This question tests the geometric properties of matrix transformations and the relationship between a matrix and its cofactor matrix.

The primary trap lies in the sign convention of the determinant (Concept 228).

Students often compute the magnitude of the volume scaling factor ( $4^3 = 64$ ) but forget that a reflection reverses orientation, making $\det(A) = -64$ .

This missing negative sign leads directly to the incorrect option $-64$ .

Furthermore, the problem elegantly bypasses brute-force computation by projecting the result onto the normal vector of the reflection plane, rewarding a deep understanding of linear algebra and vector calculus over tedious matrix multiplication.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Coordinate Axis Projections
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free