# Determinant of an orthogonal matrix

1. Sep 21, 2009

### 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(QTQ)
= det(QT)det(Q) = ((det(Q))2
and if so, what is it for -1.
Thanks.

2. Sep 21, 2009

### squenshl

Never mind, got it.

3. Sep 21, 2009

### 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!

4. Sep 21, 2009

### squenshl

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