The title is pretty much the question, but to state it more explicitly:
If we construct a square matrix A to have rows that are orthogonal unit vectors, then $AA^{T}=I$, but how can we prove that $A^{T}A=I$ ?
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$$AA^T=I$$
implies that $A^T=A^{-1}$.
Hence $$A^TA=A^{-1}A=I$$
Siong Thye Goh
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Your first sentence part is wrong. To prove that $A^{T} = A^{-1}$, you must prove both that $AA^{T}=I$ and that $A^{T}A=I$. The second condition is exactly what I've asked how to prove. – Jerry Guern Nov 12 '17 at 08:32
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Check out here for $21$ proofs of $AB=I$ then $BA=I$ which is basically left inverse is equal to the right inverse. – Siong Thye Goh Nov 12 '17 at 08:35
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Ahah! Thank you, that was the missing piece. – Jerry Guern Nov 12 '17 at 08:41