Statistics Solutions

(See Steps) Product of rotation matrices. Let A be the 2 * 2 matrix that corresponds to rotation by θ radians, defined as

Question: Product of rotation matrices. Let \(A\) be the \(2 \times 2\) matrix that corresponds to rotation by \(\theta\) radians, defined as

\[A=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\]

and let \(B\) be the \(2 \times 2\) matrix that corresponds to rotation by \(\omega\) radians. Show that $A B$ is also a rotation matrix, and give the angle by which it rotates vectors. Verify that \(A B=B A\) in this case, and give a simple English explanation.

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