A gypsy's crystal ball has a refractive index of 1.50 and diameter of 8 inch.

1)by the matrix approuch, detrermine the location of its principal points.
2)where will sunlight be focused by the crystal ball?

ray matrix for curved dialectric medium:

[1                     0    ]
[                           ]
[(n1/n2 - 1)/r         n1/n2] 

convex r>0, concave, r<0

ray matrix for travel through homogenous medium:

[1 d]
[0 1]

We have first a convex interface with n1=1, n2=n, then travel for a
distance of 2r, then a concave interface with n1=n, n2=1.  The
matrices go in reverse order, and when we plug in and multiply, the
ray matrix for the ball is:

[2/n - 1       2r/n   ]
[                     ]
[(2/n - 2)/r   2/n - 1] 

Plugging in r=4 and n=1.5, that is

1/6*[2 32]
    [-1 2]

The ray vector for a sunbeam a distance D from the axis is [D]
                                                           [0]

Run this through the ray matrix and you get 1/6*[2D]
                                                [-D]

From that you find that the focal point (where sunlight is focussed)
is 1/(3tan(1/6)) from the surface of the ball, or 4 + 1/(3tan(1/6))
from the center.  That's about 1.98 inches from the surface, or 5.98
inches from the center.