Question Bank • QB_SUB_EM_0006 NAT • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Linear and orthogonal transformations

Question Statement:

Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be a counter-clockwise rotation by an angle $\theta = 45^\circ$. If the transformation is represented by matrix $R$, then the determinant $\det(R)$ is:

Answer Key:

1.0

Step-by-Step Solution & Explanation

A 2D rotation matrix is given by $R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$. Its determinant is $\det(R) = \cos^2\theta + \sin^2\theta = 1.00$ for any angle $\theta$.
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