What is the principal moment of inertia tensor for a lamina?

[LeoChan]

Homework Statement

A thin uniform rectangular plate (lamina) is of mass m and dimensions 2a by a. Choose a coordinate system Oxyz such that the plate lies in the xy plane with origin at a corner, the long dimension being along the x-axis.

(a) The moment of inertia tensor about the origin
(b) The principal moments of inertia about the origin

Relevant Equations

Inertia tensor

The moment of inertia tensor I found out is
(1/3) (-1/2) 0
(-1/2) (4/3) 0
0 0 (5/3)

The principal moment of inertia tensor I found out is
(1/3-I) . (-1/2) . 0
(-1/2) . (4/3-I) . 0
0 . 0 . (5/3-I)

det of principal of moment inertia = 0
So (1/3-I) (4/3-I)(5/3-I)-(-1/2)(-1/2)(5/3-I)=0
I=5/3, (5+√18)/6, (5-√18)/6

The answer look so weird to me and differ from common lamina moment of inertia

 

[TSny]

Looks OK to me, except that the mass $m$ and factors of $a$ should be included.

 

[LeoChan]

Looks OK to me, except that the mass $m$ and factors of $a$ should be included.

Thanks. But just didn’t expect the answer to be irrational for a lamina, it made me so confused.