According to the theorem of perpendicular axis, the moment of inertia for any plane body for either of its axes direction normal to the plane is equivalent to the total of the moment of inertia along the two normal axis in the plane of the body that intersect the first axis in the plane.

From the figure moment of inertia about an axis passing through the centre $O$ and perpendicular to the plane of the plate is $I=I_2+I_4$.

Since the plate is rectangular of uniform thickness the $x-$axis and $y-$axis will not be symmetrical, i.e, $AC$ and $BD$ are not perpendicular to each other and this gives $I_1\ne I_3$.