Do all orthogonal matrices satisfy both $Q^TQ = I$ and $QQ^T = I$?

Question

Some definitions of orthogonal matrix (for instance, on Wikipedia) use the definition:

$$ Q^TQ=QQ^T=I $$

While other definitions (for instance, on Wolfram MathWorld) only include one of the products:

$$ QQ^T = I $$

I remember reading that $Q^TQ$ implies $QQ^T$ for orthogonal matrices, which would explain why some definitions don't list both of the products, but it is not immediately obvious to me why this is the case. From what I understand, an orthogonal matrix is a matrix whose columns are orthogonal unit vectors (i.e. $||z_i||_2^2 = 1$ and $z_i \cdot z_j = 0$ for $i \neq j$), which directly leads to the identity $Q^TQ = I$. However, I don't see why $QQ^T = I$ is necessarily true.

Answer 1

Remember that if $AB = I$ then $B=A^{-1}$ and so $BA = I$.

So if $QQ^T = I$ then $Q^T = Q^{-1}$ and so $Q^TQ=I$.

Answer 2

For an orthogonal matrix $Q$ we have

$$Q^{-1}=Q^T \iff QQ^T=Q^TQ=I$$