Determinant of the matrix of an orthogonal operator.

Question

Let $V$ be a real vector space with inner product and $\dim(V)=n$. Let $T$ be an orthogonal operator on $V$. If $A$ is the matrix of $T$ relative to the basis $\beta=\{ \beta_1,\ldots, \beta_n\}$ of $V$, prove that $\det(A)=\pm 1$

I think that if we prove that $A$ is an orthogonal matrix then the answer is clearly but $\beta$ is not an orthogonal basis of V so I don't know how can I prove that $\det(A)$ is $\pm 1$, REMEMBER THAT $\beta$ is any basis of $V$ and is not orthonormal.

Answer 1

Since $T$ is an orthogonal operator, the determinant of its matrix with respect to an orthonormal basis is $\pm1$. But the determinant is independent of the choice of the basis, and so…