stackoverflow proof sketch

Let C = 99^2 and A = {a_i,j where 0<i<99, 0<j<99} and assume that each element
in A is labeled A_n where 0<n<C. The same goes for b and B.

The sum that you are asking for, let's call it S, can be written as

S = (A_0 - B_0) + (A_0 - B_1) + ... + (A_0 - B_C) +
    (A_1 - B_0) + ...               + (A_1 - B_C) +
    ...
    (A_C - B_0) + ...               + (A_C - B_C)

A_0 occurs C times in the first row and the B terms can be written
as -(B_0 + B_1 + ... + B_C), i.e.:

C*A_0 - (B_0 + B_1 + ... + B_C)

The sum B_0 + B_1 + ... + B_C is of course the sum of all elements in B. Let's
call this S_B, and the sum of all elements in A S_A. The first row can be
written as:

C*A_0 - S_B

The second row has exactly the same B terms but has A_1 instead of A_0, so:

C*A_1 - S_B

Now we can write S like:

S = C*A_0 - S_B +
    C*A_1 - S_B +
    ...
    C*A_C - S_B

The sum of the A terms in S can be written as C*(A_0 + A_1 + ... + A_C) and S_B
occurs C times as well, so we end up with:

S = C*S_A - C*S_B
  = C(S_A - S_B)
  = 99^2 * ((A_0 + A_1 + ... + A_C) - (B_0 + B_1 + ... + B_C))