Start Learning Free

Before You Begin

Question #68001

Difficulty: 8/10 | 🟢 Free

The Question

Let Px,Py,PzP_x, P_y, P_z be the 3×33 \times 3 matrices representing projection operators onto the mutually orthogonal coordinate axes x,y,zx, y, z respectively. Define the transformation matrix T=Px+Py−PzT = P_x + P_y - P_z. Let a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} be three non-coplanar vectors. Let MM be the 3×33 \times 3 matrix whose columns are a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c}, and NN be the 3×33 \times 3 matrix whose rows are Ta⃗,Tb⃗,Tc⃗T\vec{a}, T\vec{b}, T\vec{c}. If V⃗=(Ta⃗×Tb⃗)×(Tb⃗×Tc⃗)\vec{V} = (T\vec{a} \times T\vec{b}) \times (T\vec{b} \times T\vec{c}), which of the following expressions is identically equal to det⁡(MTN)Tb⃗\det(M^T N) T\vec{b}?

Options

  • A.

    det⁡(M)V⃗\det(M) \vec{V}

  • B.

    −det⁡(M)V⃗-\det(M) \vec{V}

  • C.

    det⁡(M)2V⃗\det(M)^2 \vec{V}

  • D.

    −det⁡(M)2V⃗-\det(M)^2 \vec{V}

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

Properties of Transpose Determinant|Vector Triple Product Expansion|Coordinate Axis Projections

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the intersection of linear transformations, determinant properties, and vector triple products.

The core trap lies in the geometric interpretation of the transformation T=Px+Py−PzT = P_x + P_y - P_z.

Students often mechanically apply projection properties but fail to recognize that TT represents a reflection across the xyxy-plane, meaning its determinant is −1-1.

Consequently, the scalar triple product of the transformed vectors undergoes an orientation reversal (handedness flip), yielding [Ta⃗ Tb⃗ Tc⃗]=−det⁡(M)[T\vec{a} \ T\vec{b} \ T\vec{c}] = -\det(M).

Furthermore, constructing matrix NN from the rows of transformed vectors requires careful application of the transpose determinant property det⁡(N)=det⁡((TM)T)=−det⁡(M)\det(N) = \det((TM)^T) = -\det(M).

Failing to account for the negative sign from the reflection matrix will lead to an off-by-sign error in both the VTP expansion and the determinant product, trapping students into choosing the incorrect sign variant.

Quick Facts About This Question

Question Information

SubjectMathematics
ChapterDeterminants - Class 12
ConceptProperties of Transpose Determinant
Difficulty8
Cognitive Load7
Target Solve Time160 sec
AccessFree