How to check if a matrix is unitary?

Question

If I have the matrix for example:

$$A=\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}$$

So we need to check whether $A A^* = A^* A = I$. Is it right?

Answer 1

Yes. Alternatively (in the strictly non-complex case) you can check that $x \mapsto Ax$ is an isometry in the euclidian norm, the columns (or rows) form an orthonormal basis of $\mathbb{R}^2$ or that $A^*$ is unitary.