If a square matrix's column vectors and row vectors all have norm 1, then is the matrix orthonormal?

Question

Well, this is a question I had to ask myself while solving a problem that asked me to prove a matrix is orthonormal.

I could show that both the column vectors and the row vectors of said matrix all had unit length, but didn't know how to proceed from there, as I could also not find a way to directly show the orthogonality of the column vectors.

In the end, I tried figuring out a way to prove whether or not the statement in the title followed or not, to no avail. Is it true, or is there a counterexample?

Answer 1

$$ \pmatrix{s & s\\ s & s} $$ where $s = \frac{\sqrt{2}}{2}.$

Answer 2

If $O=(o_{j,k})$ is almost any $n\times n$ orthogonal matrix with $n>1$ then $A=(|o_{j,k}|)$ is a counterexample.

Answer 3

In fact, one can find counterexamples like this one:

$$M=\begin{pmatrix}s&s&u&u\\s&-s&u&u\\u&u&-s&s\\u&u&s&-s\end{pmatrix}$$

where $s=0.7$ and $u=0.1$ with moreover $\det(M) \ne 0$.