Orthogonal matrices proof Let $\upsilon _{n} $ be the set of all $n \times  n$ orthogonal matrices(for all fixed $n$).
Show that $\upsilon _{n} $ is not a subspace of $\ M _{n\times n} $. Thank you !
Additional:
Suppose $A \in \upsilon_{n}$, what are the possible values for $detA?^{11}$ and possible engenvalues for  $A?^{12}$. 
I am really confused about the additional one. Thanks again.  
 A: Hint: Is $0\in \upsilon_n{}{}{}{}{}{}{}$?
A: The set of $n \times n$ orthogonal matrices is not closed under scalar multiplication, since for any scalar $c$, and given an orthogonal matrix $A$, we have
$$(cA)(cA)^T = c^2AA^T = c^2I$$
which isn't the identity matrix if $c \ne \pm 1$.  Or more simply, using Alex's hint, observe that the $0$ matrix is not orthogonal, but every subspace must contain $0$.
A: Hint: Is the sum of two orthogonal matrices is an orthogonal matrix? Check the following example
$$ \pmatrix{1&0\\ 0&1}+\pmatrix{0 & 1 \\ 1 & 0}=\pmatrix{1&1 \\ 1&1 }. $$
Clearly the new matrix is not orthogonal.
A: Hint to additional question: if $A$ is orthogonal, it is very easy to show $A^T A = I$.  In fact, this is equivalent to being orthogonal.  More hints: what is $\operatorname{det}(BC)$ if $B$ and $C$ are square?  What is $\operatorname{det}(A^T)$ in terms of $\operatorname{det}(A)$?
For the eigenvalues, assume $A\mathbf{x} = \lambda\mathbf{x}$, $\mathbf{x} \neq \mathbf{0}$.  Calculate the norm squared of both sides, using $\|\mathbf{v}\|^2 = \mathbf{v}^H \mathbf{v}$.  Here "$H$" is conjugate transpose, which you need if you are looking for complex eigenvalues and not just real ones.
