Determinant of an orthogonal matrix

squenshl · Sep 21, 2009

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

squenshl · Sep 21, 2009

Never mind, got it.

daviddoria · Sep 21, 2009

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 · Sep 21, 2009

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

Determinant of an orthogonal matrix

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