What are the possible values of the determinant of an orthogonal matrix?

[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?

 

[cristo]

What is the definition of an orthogonal matrix?

 

[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 I am not sure how to use this to find out all values of the determinant

 

[cristo]

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?

 

[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 I am not understanding how that is = det (M)

 

[cristo]

and I do know the det(M^t) = det (M)

but det (M^tM) = 1 and I am 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?

 

[mjsd]

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

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

 

[salman213]

oh okk..cool..thanks