I'm computing the principal axes and moments of a triangle at its centroid. The triangle vertices are at the points $(12,0,0)$, $(0,24,0)$, $(0,0,36)$. According to my calculations, the relevant characteristic equation is $\lambda^3 - 112\lambda^2 + 2352\lambda = 0$. This has roots 84, 28, 0. It's the zero eigenvalue that's bothering me. The values 84 and 28 are moments of inertia about axes lying in the plane of the triangle. So, according to the perpendicular axis theorem, the moment of inertia around the axis perpendicular to this plane ought to be $84 + 28 = 112$. So, why is $112$ not a root of the characteristic polynomial ?? Puzzled.
If it helps -- the characteristic polynomial is derived from the matrix (the so-called "inertia tensor") whose rows are $[104,8,12]$, $[8,80,24]$, $[12,24,40]$. It's symmetric, obviously.