In which cases is the inverse of a matrix equal to its transpose?

Question

In which cases is the inverse of a matrix equal to its transpose, that is, when do we have $A^{-1} = A^{T}$? Is it when $A$ is orthogonal?

Answer 1

If $A^{-1}=A^T$, then $A^TA=I$. This means that each column has unit length and is perpendicular to every other column. That means it is an orthonormal matrix.

Answer 2

You're right. This is the definition of orthogonal matrix.