Given the matrix $A = 1/7\begin{bmatrix} 6 & 3 & a_{13} & \\ -2 & 6 & a_{23} \\3 & a_{32} & a_{33} \end{bmatrix}$
How do replace the $a_{ij}$'s with real entries such that the matrix becomes a rotation matrix?
I know it should have a determinant of - 1 and all eigenvalues should be either 1 or -1, but it turns out to be a four variables equation, how do I find those values?
Edit: I forgot the scalar in front of the matrix.
Each column has to have length $1$, so $$3^3+6^2+a_{32}^2=7^2.$$ This determines $a_{32}$ up to sign. But the first two columns must be orthogonal, and that will determine the sign of $a_{32}$.
Also the rows have to have length $1$. Therefore $$6^3+3^2+a_{13}^2=7^2$$ etc. So one gets the entries in the last column up to sign. Orthogonality of rows means that one sign determines the others. Finally a rotation matrix has determinant $1$, so this determines the final sign.