# Answer to Question #2665 in Linear Algebra for nafeh

Question #2665

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.

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.

Expert's answer

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.

& 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.

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