Matrix that represents a stretching along $X$-axis by a factor $2$ followed by a clockwise rotation of $60$ degrees.

Question

I was given to find a matrix that represents a stretching along $X$-axis by a factor $2$ followed by a clockwise rotation of $60$ degrees.

Basically I have to translate and rotate a point.

So I can use the following matrix.

$\begin{bmatrix} 2\cos(2\pi – \frac{\pi}{3}) & 0\\
0 & \cos(2\pi – \frac{\pi}{3}) \end{bmatrix}$

Have I gone wrong anywhere?

Answer 1

Short answer is no - you need to go about this in two steps

1. The stretch - This can be defined the matrix

$$M_S = \begin{bmatrix}2 & 0 \\ 0 & 1\end{bmatrix}$$

2. The rotation - This is a standard rotation matrix

$$M_R = \begin{bmatrix}\cos (- \frac{\pi}{3}) & -\sin(-\frac{\pi}{3}) \\ \sin(-\frac{\pi}{3}) & \cos(-\frac{\pi}{3})\end{bmatrix}$$

Now your transformation would be described as

$$T = M_RM_S$$