Choose the number x to maximize the determinant (this means maximizing the
entropy!) of the symmetric matrix
S = 4 2 x
2 4 2
x 2 4
How exactly are you supposed to do this without guessing?
Find the determinant of S. A trick in high school I learned gives me det(S) = [4*4*4 + 2*2*x + x*2*2] - [x*4*x + 2*2*4 + 4*2*2].
Then det(S) = [64 + 4x + 4x] - [4x^2 + 16 + 16]
Then det(S) = 64 + 8x - 4x^2 - 32
Then det(S) = 32 + 8x - 4x^2
Remember the maximum of ax^2 + bx + c = 0 where a < 0 occurs when x = -b/2a.
The maximum det(S) occurs when x = -8/(2*-4) = -8/-8 = 1.
Thus, x = 1.