Does positive-definite order imply determinant order?

[hadron23]

Hi,

Given two real-valued positive-definite matrices A and B, assume one is greater than the other with respect to positive definite ordering. That is, A>B. Does the following implication hold?

$$A>B \Rightarrow \text{det}(A)>\text{det}(B)$$

Thanks.

 

[arkajad]

Interesting question. I checked for 2x2 matrices and in this case it seems to hold. But for nxn?

 

[hadron23]

I guess the a sufficient condition for this is if $$A>B$$ then do all the eigenvalues of $$A$$ dominate all of the eigenvalues of $$B$$?