Determinant of an orthogonal matrix

[squenshl]

How is it the determinant of an orthogonal matrix is $$\pm1$$.
Is it:
Suppose Q is an orthogonal matrix $$\Rightarrow$$ 1 = det(I) = det(Q^TQ)
= det(Q^T)det(Q) = ((det(Q))²
and if so, what is it for -1.
Thanks.

 

[squenshl]

Never mind, got it.

 

[daviddoria]

Maybe you can share your new knowledge with the forum so that when other users see the question they can also see the answer!

 

[squenshl]

Sweet.
Suppose Q is orthogonal, then A^TA = I_n
$$\Rightarrow$$ det(A)det(A^TA) = det(A)det(A) = (det(A))² = 1
Which implies that det(A) = $$\pm\sqrt{1}$$ = $$\pm1$$