Exercise 3.4.4. Given two orthogonal matrices and , show that their product is also orthogonal. If represents rotation through the angle and represents rotation through the angle , what does represent? What trigonometric identities for and can be found in multiplying and ?

Answer: For any orthogonal matrix we have . We then have

Since the matrix is orthogonal.

If the matrices and represent rotations through and respectively, then represents rotation through , and will contain elements including and . These will be produced in the matrix multiplication through the following trigonometric identities:

where and come from   and and come from .

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.