Determining all scalars $a \in \mathbb{R}$ for which a matrixrepresentation is orthogonal?

Question

Problem: Let $a \in \mathbb{R}$ and \begin{align*} T: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}: A \mapsto aA. \end{align*} Determine all $a \in \mathbb{R}$ for which the matrix of $T$ with respect to the standard basis is an orthogonal matrix.

I have no idea how to do this. I know that $A^T = A^{-1}$ or $A^T \cdot A = I_n$ for an orthogonal matrix, but how can I use this here?

Answer 1

Hint The matrix associated to $T$ is $a$ times the identity matrix $I$, where $I$ is the identity matrix acting on $\Bbb R^{n\times n}$.

Answer 2

Hint: And orthogonal transform maps unit vectors to unit vectors