Question

Consider a plane and a line given by equation ax + by = 0. Write a 2×2 matrix, M, of the mirror reflection (flip) around this line. Calculate det(M) and M² and discuss the results.

Answer 1

In slope intercept form, the ax + by = 0 can be written as y = -(a/b)x.

The matrix representing reflection across the line y = mx is A =

-(m2-1)/ (m2+1)	2m/(m2+1)
2m/(m2+1)	(m2-1)/(m2+1)

Here, m = -a/b so that M =

-(a2-b2)/ (a2+b2)	-2ab/(a2+b2)
-2ab/(a2+b2)	(a2-b2)/ (a2+b2)

Then, det(M) = -(a²-b²)²/(a²+b²)²- 4a²b²/(a²+b²)² = -(a⁴-2a²b²+b⁴+4a² b²)/ (a²+b²)² = -(a⁴+2a²b²+b⁴)/ (a²+b²)² = -(a²+b²)²/(a²+b²)² = -1.

Also, M² = M.M =

1	0
0	1

The matrix M² is the identity matrix I , as it reflects a second time across the line ax+by = 0, thereby undoing the first reflection across this line.