matrix of a rotation about the origin

Question

I am having problems with a question on my homework assignment for my linear algebra class:

The transformation matrices of two reflections over two lines L1,L2,in the plane are M1 and M2. Is it possible that M3=M1+M2 is the matrix of a rotation about the origin?

Answer 1

Write a general matrix $M_1$ for a reflection about a line. If you can't remember how to write the elements of such a matrix, here are a couple of reminders: Reflect point across line with matrix, Reflection across a line?.

Write another general matrix $M_2$ for a reflection about a line, not necessarily the same one. Add the two matrices. Now you have a matrix $M_3.$

Write a general matrix for a rotation. Call it $M_4.$

You should now have two matrices, $M_3$ and $M_4,$ each of which has four entries, each entry is an expression in one or two variables (one variable for the rotation, two for the sum of the reflection matrices).

Look for patterns. A reflection matrix has a very specific pattern of elements that are equal to each other or exact negatives of each other. This affects the relationships that can exist among the elements of $M_3$. A rotation matrix also has a pattern which must exist in $M_4$.

If the answer to the question is yes, there must be a way to get the two patterns to match. You would get that answer by finding values of the variables that make $M_3 = M_4$. If it is impossible to match the two patterns, the answer is no.

Answer 2

You can prove this using some general properties of rotations and reflections in 2-D, and of determinants. Rotations are isometries (length and angle-preserving transformation) that also preserve orientation, so their determinant is always equal to $1$. Similarly, reflections are orientation-reversing isometries, with determinant $-1$. The composition of two isometries is clearly also an isometry, and the determinant of a product is the product of the determinants, so the product of two reflections is an isometry with determinant $1$—a rotation. (There are other orientation preserving isometries, but they’re not linear transformations.)