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?

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- Apr 3rd 2013, 08:36 PMkkarMaximazing determinant of symmetric matrix
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?

- Apr 3rd 2013, 08:38 PMGusbobRe: Maximazing determinant of symmetric matrix
- Apr 5th 2013, 10:51 AMmathguy25Re: Maximazing determinant of symmetric matrix
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. - Apr 5th 2013, 02:02 PMkkarRe: Maximazing determinant of symmetric matrix
That makes total sense. Thanks!