A $3 \times 3$ matrix with unit row and unit column but all unique elements must be orthogonal matrix.

Question

I am using basic matrix theory to prove that if a $3 \times 3$ matrix has unit rows and unit columns but all entries are unique then matrix must be orthogonal. But I have trouble proving this. All I have is $9$ equations with $9$ unknowns. Is there any other way to prove this?

Answer 1

You cannot prove it because it is false. Repeatedly generate a random, entrywise positive doubly stochastic matrix $S$ until it has distinct elements. Then take its entrywise square root $R$. E.g. $$ S=\frac{1}{30}\pmatrix{9&17&4\\ 10&6&14\\ 11&7&12} \implies R=\frac{1}{\sqrt{30}}\pmatrix{3&\sqrt{17}&2\\ \sqrt{10}&\sqrt{6}&\sqrt{14}\\ \sqrt{11}&\sqrt{7}&\sqrt{12}}. $$ By construction, all rows and columns of $R$ are unit vectors and all elements of $R$ are distinct, but $R$ cannot possibly be orthogonal because it is entrywise positive.

(In the above, if one allows taking negative square roots of some elements of $S$, then sometimes the resulting matrix $R$ is orthogonal. The matrix $S$ in this case is said to be orthostochastic.)

Alternatively, generate a random orthogonal matrix $U$ repeatedly until its elements have distinct absolute values and it has not any zero element on the first row or the first column. Then flip the sign of the first element of $U$ to obtain a new matrix $V$. E.g. $$ U=\pmatrix{\frac{1}{\sqrt{35}}&\frac{7}{\sqrt{234}}&\frac{79}{\sqrt{8190}}\\ \frac{3}{\sqrt{35}}&\frac{11}{\sqrt{234}}&-\frac{43}{\sqrt{8190}}\\ \frac{5}{\sqrt{35}}&-\frac{8}{\sqrt{234}}&\frac{10}{\sqrt{8190}}} \implies V=\pmatrix{\color{red}{-\frac{1}{\sqrt{35}}}&\frac{7}{\sqrt{234}}&\frac{79}{\sqrt{8190}}\\ \frac{3}{\sqrt{35}}&\frac{11}{\sqrt{234}}&-\frac{43}{\sqrt{8190}}\\ \frac{5}{\sqrt{35}}&-\frac{8}{\sqrt{234}}&\frac{10}{\sqrt{8190}}}. $$ The rows and columns of $V$ will then be unit vectors and all elements of $V$ are distinct, but $V$ by construction cannot be orthogonal.