Determinant of $U$, Determinant of $U^T$

Question

Given an $n\times n$ matrix $U$ such that $U^TU = I_n$, the $n\times n$ identity matrix.

Then what are the possible values of the determinant of $U$?

Answer 1

I am writing the answer through an ordinary mobile. So please excuse me with the proper formatting. . . Determinant of the product $AB$ is equal to the product of the determinants of $A$ and $B$. Also determinant of the transpose is same as determinant of the matrix. So the answer is $\pm 1$

Answer 2

$U^T\cdot U=I_n$ means your matrix is an orthogonal matrix (http://en.wikipedia.org/wiki/Orthogonal_matrix)

Hence it's determinant is $\pm 1$ :

---

proof :

$\det (U)²=\det (U)\det (U^T)$ because $\det (U)=\det(U^T)$

Therefore $\det (U)^2=\det(U^T\cdot U)=\det (I_n)=1$