In this lesson, we will learn how to find the matrix of two or more consecutive linear transformations.

Q1:

Let the matrix 𝐴 represent rotation in the plane through an angle of 𝜃 and let the matrix 𝐵 represent reflection in the 𝑥-axis.

What is the matrix 𝐴?

What is the matrix 𝐵?

What is the matrix 𝐴𝐵?

Q2:

Suppose that the matrix 𝐴 represents rotation about the origin through an angle of 𝜃 (measuring between 0∘ and 90∘) and the matrix 𝐵 represents reflection in the 𝑥-axis.

Find 𝑀=𝐴𝐵.

Note that 𝑀 is a reflection in a line through the origin. Let this line of reflection have equation 𝑦=𝑘𝑥. By considering the image of the vector ⟨1,0⟩, determine the measure of the angle between the 𝑥-axis and the line of reflection.

What, therefore, is the slope 𝑘 of the line of reflection?

If v lies in the direction of the line of reflection, what is 𝑀v?

By solving the equation obtained in the previous part, find another expression for the slope 𝑘 of the line of reflection.

Q3:

Suppose 𝐴 and 𝐵 are 2×2 matrices, with 𝐴 representing a counterclockwise rotation of 30∘ about the origin and 𝐵 representing a reflection in the 𝑥-axis. What does the matrix 𝐵𝐴 represent?

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