Answered: Let R be an n × n plane rotation. What…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let R be an n × n plane rotation. What is the
value of det(R)? Show that R is not an elementary
orthogonal matrix.

Expert Solution

Step 1: Given :

$R$ is a $n \times n$ plane rotation.

Step 2: Calculation of the determinant of R.

Since $R$ is a plane rotation, it is a linear transformation that rotates all vectors in the plane by the same angle.
Let $θ$ be the angle of rotation. Then $R$ can be expressed as :

$R = [\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}]$

To compute the determinant of $R$ , we can use the formula for the determinant of a $2 \times 2$ matrix :

$det (R) = |\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}| = \cos^{2} θ + \sin^{2} θ = 1$

Therefore, the determinant of $R$ is always equal to $1$ .

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