Question

Find the determinant value of an orthogonal matrix.

Answer 1

Let "A" be an orthogonal matrix.
Then by definition of an orthogonal matrix, the transpose of A is equal to the inverse of A:
A^T = A^(-1)

Then remember what the definition of an inverse matrix is:
A*A^(-1) = I

"I" will be the identity matrix. Now you can substitute in A^(T) for A^(-1) since A^(T) = A^(-1):
A*A^(T) = I

Take the determinant of both sides:
det(A*A^(T)) = det(I)

Remember that the det(I) is always 1:
det(A*A^(T)) = 1

Then you should also remember a theorem det(A*B) = det(A) * det(B):
det(A) * det(A^(T)) = 1

And finally there's one more theorem you need to know: that det(A) = det(A^T):
det(A) * det(A) = 1

Simplify:
(det(A))² = 1

And you get:
det(A) = ±1