Recall that P is a projection matrix if and only if PT=P and P2=P. Also, recall that R is a reflection matrix if and only if RT=R and R2=I. Finally, recall that A is an orthogonal matrix if and only if AAT=I=ATA.
Let R be a reflection matrix. Determine whether each of the following statements is true or false.
1. If R is invertible, then R=I.

2. I−R is a projection matrix.

3. R−I is a projection matrix.

4. det(R)=1 or det(R)=−1.

5. If S is also a projection matrix, then so is R+S.

6. 1 is an eigenvalue for R.

Question

Recall that P is a projection matrix if and only if PT=P and P2=P. Also, recall that R is a reflection matrix if and only if RT=R and R2=I. Finally, recall that A is an orthogonal matrix if and only if AAT=I=ATA. 
Let R be a reflection matrix. Determine whether each of the following statements is true or false.
  1. If R is invertible, then R=I.

  2. I−R is a projection matrix.

  3. R−I is a projection matrix.

  4. det(R)=1 or det(R)=−1.

  5. If S is also a projection matrix, then so is R+S.

  6. 1 is an eigenvalue for R.

anonymous · Answer

help guys

zarkon · Answer

http://openstudy.com/study#/updates/4fdbf607e4b0f2662fd226b2

anonymous · Answer

one of the answers wasnt right

zarkon · Answer

6 is false..does that help

anonymous · Answer

well I am sure it is true

anonymous · Answer

I am not sure about 4

zarkon · Answer

$$R=\left[\begin{matrix}-1& 0 \ 0 & -1\end{matrix}\right]$$

anonymous · Answer

this is not a reflection

zarkon · Answer

R'=R
R^2=I

but the only eigenvalues are -1 and -1  (repeated eigenvalue)

zarkon · Answer

why not...I'm using your definition

anonymous · Answer

it is a rotation matrix

zarkon · Answer

"recall that R is a reflection matrix if and only if RT=R and R2=I"

the matrix above satisfies both of your criteria

anonymous · Answer

yes even the rotation matrices satisfy that but the diagonal of a certain matrix has the same entries then it is a rotation

zarkon · Answer

this is a double reflection...it is a reflection matrix(as per your own definition) 

 (you can also consider it a rotation matrix too)

anonymous · Answer

double reflection is a rotation

zarkon · Answer

then I can't help you...I gave you a matrix that satisfied your definition, but you are unable or unwilling to believe it.

anonymous · Answer

not me it is what we learned is unable to accept that

zarkon · Answer

the matrix I gave you satisfies the two critera of your definition does it not?

anonymous · Answer

the thing is that the rotation and the reflection are like brothers same mother same mother but they are different anyway thanks a lot for ur help. :)

anonymous · Answer

same father

zarkon · Answer

what is your hang up?...what about the identity matrix.  it is a reflection and rotation matrix

anonymous · Answer

I is a rotation and reflection or u can say it is nothing

zarkon · Answer

what I gave you above follows your definition.  If you don't like that fact then you need to find a new definition.

anonymous · Answer

thanks a million, ur help is appreciated :)