Operations with orthogonal matrices

Question

I have the following problem in linear algebra:

Let $n = 1,3,5,7,9,\dots$ and an orthoginal matrix $C$ (with dimensions $n\times n$) with $\det C=1$. Prove that:

$$C^T (C-I) = (I-C) ^T$$

I am aware of that for any orthogonal matrix, the transpose matrix is equal to the inverse one, but still I face difficulty in proving the above. I would appreciate your quidance.

Thank you very much in advance.

Answer 1

For any orthogonal matrix $C$,

$$C^T(C-I)=C^TC-C^T=I-C^T=(I-C)^T$$