Yes, every orthogonal matrix has a determinant.
Understanding Orthogonal Matrices
An orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors. This means that each vector has a length of 1 (normalized), and they are all perpendicular to each other.
Determinants of Orthogonal Matrices
The determinant of an orthogonal matrix is always either +1 or -1. This property is directly linked to the orthonormality of its vectors.
- Determinant = +1: The orthogonal matrix represents a rotation.
- Determinant = -1: The orthogonal matrix represents a reflection.
Example
Consider the following orthogonal matrix:
[ 0 1 ]
[ -1 0 ] This matrix represents a rotation of 90 degrees counter-clockwise. Its determinant is:
(0 0) - (1 -1) = 1
Conclusion
Orthogonal matrices always have a determinant, and this determinant is either +1 or -1. This property is fundamental to understanding the geometric transformations represented by orthogonal matrices.
