An \(n \times n\) matrix \(A\) over a field \(F\) is called orthogonal if \(AA^t = I\). If \(A\) is orthogonal, show that \(\det A = \pm 1\). Give an example of an orthogonal matrix for which \(\det A = -1\).
Solution
From \(AA^t=I\) and determinant multiplicativity, \[ 1=\det I=\det(A)\det(A^t)=(\det A)^2. \] Hence \(\det A=\pm1\).
For example, \[ A=\operatorname{diag}(-1,1,\ldots,1) \] is orthogonal and has determinant \(-1\).