$3\times 3$ orthogonal matrix, which doesn't consist of zeros and ones

Question

I'm stuck with my homework in a subject called Matrices in Statistics. Can you guys help with the following task? I would be very thankful!

The task is as follows:

Find a $3\times 3$ orthogonal matrix, which doesn't consist of zeros and ones. A matrix, for which no term is $0$ or $1$. Orthogonality is need to be shown as well!

Answer 1

You can take$$\begin{bmatrix}\frac35&-\frac45&0\\\frac45&\frac35&0\\0&0&1\end{bmatrix}.$$

If you wan an orthonormal matrix withou any $0$ and without any $1$, you can use$$\begin{bmatrix}-\frac{3}{125} & -\frac{96}{125} & \frac{16}{25} \\ \frac{96}{125} & -\frac{53}{125} & -\frac{12}{25} \\ \frac{16}{25} & \frac{12}{25} & \frac{3}{5}\end{bmatrix}.$$

Answer 2

Dealing with these type of questions using vectors makes it a lot easier. So basically, we have three vectors say a, b and c(sorry for not using proper notations). The direction ratios of these vectors are the elements of the matrix. We get, a.b=b.c=c.d=0(mutually perpendicular vectors) and mod(a)=mod(b)=mod(c)=1(unit vectors). So, we have to look for three mutually perpendicular unit vectors without 0 as any of its direction ratios. Take x, y and z axes and rotate them by a certain angle to get the answer. (Doing it on your own will be better)