Allied & ICAR Exam Bank • ALLIED_2609 NAT • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Linear and orthogonal transformations

Question Statement:

A linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ rotates every vector counterclockwise by $\theta = 45^\circ$ and scales it by a factor of $\sqrt{2}$. The trace of the transformation matrix is:

Answer Key:

2

Step-by-Step Solution & Explanation

The matrix is $M = \sqrt{2} \begin{bmatrix} \cos 45^\circ & -\sin 45^\circ \\ \sin 45^\circ & \cos 45^\circ \end{bmatrix} = \sqrt{2} \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$. The trace is $1 + 1 = 2$.
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