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- Jan 26, 2012

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**Problem**: Let $O$ be an $n \times n$ orthogonal real matrix, i.e. $O^TO=I_n$. Prove that:

a) Any entry in $O$ is between -1 and 1.

b) If $\lambda$ is an eigenvalue of $O$ then $|\lambda|=1$

c) $\text{det O}=1 \text{ or }-1$

**Solution**: I want to preface this with that although this is a 3-part question and our rules state we should only ask one question at a time, I believe that all parts use the same concepts so it's more efficient to put them together in one thread. If that proves to be untrue, then I'll gladly split the thread.

Right now I see the solution for part (c): It uses the fact that $\text{det AB}=\text{det A }\text{det B}$.

$1= \text{det } I_n=\text{det }OO^T=\text{det }O\text{ det }O^T=\text{det }O\text{ det }O=(\text{det }O)^2$, thus $\text{det }O$ must be 1 or -1.

Any ideas on (a) and (b)?