Symmetrix and Orthognoal Matrix

[helenli89]

I have shown that 1 and 2 are equivalent but I'm stuck with the 3 point.

My solution for 1 and 2 as follows:
If A is orthogonal matrix, than AA(transpose) = I.
Since A^2 = AA = AA(transpose) since A is symmetric therefore A^2=I
If A^2=I then A^2 = AA = AA(transpose) since A is symmetrice, then AA(transpose) = I therefore A is orhogonal matrix.
But how do I show that the eigenvalues of A are +/- 1?

 

[helenli89]

I also not sure how to do question 6. But I have attmpted it. Please see attachment for questions and my attempt.
Please give your comments on the proving.
Thank you very much.

PS question 7 is for you to play with.