Answer to Question #2665 in Linear Algebra for nafeh

Question

Let A=1/3 (-2 -1 2
2 -2 1
1 2 2)
Prove that A is the product of a rotation and a reflection. Prove that A is an orthogonal matrix.

Answer 1

2) To show that A is an orthogonal matrix it is necessary to verify that
& a) the sum of squares of any row of A is equal to 1:
e.g. for the first row:
((-2/3)^2 +(-1/3)^2+(2/3)^2)=
(4+1+4)/9=1

b) the scalar product of any two distinct rows is zero, e.g for the first and second rows:
-2*2+(-1)*(-2)+2*1=0

1) Prove A is the product of a rotation and a reflection.
Due to 2) we know that A is orthogonal. If det(A)>0, then A is a rotation,& while for det(A)<0 it is a of a product of rotation and a reflection.
Thus it is necessary to show that det(A)<0.