# Determinant of Orthogonal

by salman213
Tags: determinant, orthogonal
 P: 303 Hi I had a final today and one of the questions was find all the possible values of det Q if Q is a orthogonal matrix I m still wondering how would I do this? Any ideas?
 Mentor P: 8,272 What is the definition of an orthogonal matrix?
 P: 303 well i guess the vectors which make up the matrix are orthogonal and so have a dot product of 0? and the transpose of an orthogonal matrix is its inverse but im not sure how to use this to find out all values of the determinant
Mentor
P: 8,272

## Determinant of Orthogonal

Ok, so you know the transpose of an orthogonal matrix is its inverse. So, we have $M^TM=I$. Now, let's take the determinant of this; $det(M^TM)=det(I)$. I presume you know what the right hand side is equal to. Now, what can one say about the relationship between the determinant of a matrix, and the determinant of its transpose?
 P: 303 but how is the determinant of(M^TM) = det(M) if M is a orthogonal matrix by the way since you said det (i) its 1..right? and I do know the det(M^t) = det (M) but det (M^tM) = 1 and im not understanding how that is = det (M)
Mentor
P: 8,272
 Quote by salman213 and I do know the det(M^t) = det (M) but det (M^tM) = 1 and im not understanding how that is = det (M)
Right, so putting these two facts together we have det(M2)=1. Can you find det(M) from this expression?
 HW Helper P: 858 hmmm...salman213 perhaps this is the theorem you want det(AB) = det(A)det(B)
 P: 303 oh okk..cool..thanks

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