A rigid body has principal moments of inertia \(I_{x x}=I_{0}, I_{y y}=I_{z z}=2 I_{0} / 3\). (a) Find all elements of the moment of inertia matrix in a reference frame that has been rotated by \(30^{\circ}\) about the \(z\) axis in the counterclockwise sense relative to the initial axes. (b) In this new (primed) frame the moment of inertia matrix has the form

\[\left(\begin{array}{ccc}
I_{x x}^{\prime} & I_{x y}^{\prime} & I_{x z}^{\prime} \\
I_{y x}^{\prime} & I_{y y}^{\prime} & I_{y z}^{\prime} \\
I_{z x}^{\prime} & I_{z y}^{\prime} & I_{z z}^{\prime}
\end{array}\right)\]

where the nine entries were found in part (a). Now pretending that you do not already know the answer, diagonalize this matrix to find the principal moments of inertia (That is, subtract \(I\) from each of the diagonal elements in the matrix, and then set the determinant of the resulting matrix equal to zero. This will give a cubic equation in \(I\), which when solved will give the three principal moments of inertia.