## Linear Algebra and Its Applications, Exercise 3.4.4

Exercise 3.4.4. Given two orthogonal matrices $Q_1$ and $Q_2$, show that their product $Q_1Q_2$ is also orthogonal. If $Q_1$ represents rotation through the angle $\theta$ and $Q_1$ represents rotation through the angle $\phi$, what does $Q_1Q_2$ represent? What trigonometric identities for $\sin (\theta + \phi)$ and $\cos (\theta + \phi)$ can be found in multiplying $Q_1$ and $Q_2$?

Answer: For any orthogonal matrix $Q$ we have $Q^TQ = I$. We then have $(Q_1Q_2)^T(Q_1Q2) = (Q_2^TQ_1^T)(Q_1Q2) = Q_2^T(Q_1^TQ_1)Q_2 = Q_2^TQ_2 = I$

Since $(Q_1Q_2)^T(Q_1Q2) = I$ the matrix $Q_1Q_2$ is orthogonal.

If the matrices $Q_1$ and $Q_2$ represent rotations through $\theta$ and $\phi$ respectively, then $Q_1Q_2$ represents rotation through $\theta + \phi$, and will contain elements including $\sin (\theta + \phi)$ and $\cos (\theta + \phi)$. These will be produced in the matrix multiplication through the following trigonometric identities: $\sin (\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi$ $\cos (\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi$

where $\sin \theta$ and $\cos \theta$ come from $Q_1$  and $\sin \phi$ and $\cos \phi$ come from $Q_2$.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fifth Edition and the accompanying free online course, and Dr Strang’s other books .

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