# number of n by n orthogonal matrices with integer entries

#### bombardior

are any of the following 2 statements true? what do they mean each of whose entries is an integer? does that mean all of the entries in the square matrix are all the same integers or different integers or have at least one integer? Thanks!

#### tonio

are any of the following 2 statements true? what do they mean each of whose entries is an integer? does that mean all of the entries in the square matrix are all the same integers or different integers or have at least one integer?

It means exactly what it is written: that each and all of its entries are integers numbers. It can't be the same integer all along, of course, lest the matrix is singular, which is absurd since orthogonal matrices are regular. Thanks!
I think the first statement is clearly false: we know that, for $$\displaystyle n=2$$ , a real matrix is orthogonal iff it is of the form $$\displaystyle \begin{pmatrix}\cos\theta&\!\!\!-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$$ , or $$\displaystyle \begin{pmatrix}\cos\theta&\sin\theta\\\sin\theta&\!\!\!-\cos\theta\end{pmatrix}$$ ,
for $$\displaystyle \theta\in [a,a+2\pi)\,,\,\,a\in\mathbb{R}$$ .

Choosing $$\displaystyle a=0$$ we get that the only values for which we get an integer matrix are $$\displaystyle \theta=0,\,\pi/2,\,\pi,\,3\pi/2$$ , and the matrices are:

$$\displaystyle \begin{pmatrix}1&0\\0&1\end{pmatrix}\,,\,\,\begin{pmatrix}1&0\\0&\!\!\!-1\end{pmatrix}$$

$$\displaystyle \begin{pmatrix}0&\!\!\!-1\\1&0\end{pmatrix}\,,\,\,\begin{pmatrix}0&1\\1&0\end{pmatrix}$$

$$\displaystyle \begin{pmatrix}\!\!\!-1&0\\0&\!\!\!-1\end{pmatrix}\,,\,\,\begin{pmatrix}\!\!\!-1&0\\0&1\end{pmatrix}$$

$$\displaystyle \begin{pmatrix}0&1\\\!\!\!-1&0\end{pmatrix}\,,\,\,\begin{pmatrix}0&\!\!\!-1\\\!\!\!-1&0\end{pmatrix}$$

From the example above we can see that for any $$\displaystyle n\in\mathbb{N}$$ there is only a finite number of integer orthogonal matrices since their entries must be all either $$\displaystyle \pm 1\,\,\,or\,\,\,0$$ (why?).

The final observation for you to calculate the number of such matrices is that an integer orthogonal matrix must have only one entry equal to $$\displaystyle \pm 1$$ and all the rest equal

to $$\displaystyle 0$$ in each column and each row (why? Check what'd happen with that column/row's length if there were two or more non-zero entries...(Wink))

Tonio

• bombardior