Find Determinant of A, when the Product of A and Transpose of A is Identity

Question

If $A^T . A = I$, prove that determinant A = +-1.

I don't even know where to start. Can somebody please give me a good start at least.

Answer 1

Since $\det(A)=\det(A^T)$ we have $\det(AA^T)=\det(I)=1$. This factors to $\det(A)\det(A^T)=1$ which in the real case we know means $\det(A)=\det(A^T)= \pm 1$.