# Determinant of Orthogonal

1. Dec 10, 2007

### salman213

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?

2. Dec 10, 2007

### cristo

Staff Emeritus
What is the definition of an orthogonal matrix?

3. Dec 10, 2007

### salman213

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

4. Dec 10, 2007

### cristo

Staff Emeritus
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?

5. Dec 15, 2007

### salman213

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)

6. Dec 15, 2007

### cristo

Staff Emeritus
Right, so putting these two facts together we have det(M2)=1. Can you find det(M) from this expression?

7. Dec 15, 2007

### mjsd

hmmm...salman213 perhaps this is the theorem you want

det(AB) = det(A)det(B)

8. Dec 16, 2007

### salman213

oh okk..cool..thanks