Positive Definite Matrix Determinant 
Prove that a positive definite matrix has positive determinant and
  positive trace.

In order to be a positive determinant the matrix must be regular and have pivots that are positive which is the definition. Its obvious that the determinant must be positive since that is what a positive definite is, so how can I prove that?

One more proof to show that the determinant of a +ve definite matrix  is +ve:
let A be a +ve definite matrix. Now we know that A can be written as :
A = $(V)D(V^T)$ ,which is the single value decomposition form of the +ve definite matrix A.Here $V^T$ and V are the orthogonal vectors such that $ (V^T)V $ = 1 .Thus we can also write :
$|V^T|$ |V| = 1 ---(1).
Also D is the diagonal matrix with eigen values of A as its diagonal elements.
Now, A = $(V)D(V^T)$ .
or $|A| = |V| |D| |V^T| $ .
or |A| = |D| (from (1) ) ---(2) .
|D| = product of eigen values ----(3).
Thus if we prove that all the eigen values of a +ve definite matrix are +ve then we are done.
$(x^T)A(x)$ > 0 .
Ax=bx ,where b is the eigen value. 
thus $(x^T)bx > 0 $.
$b(x^T)x > 0 $.
$(x^T)x > 0 $ (always) .
thus b>0 .
thus from (2) and (3) : 
|A| = |D| > 0 
or |A| > 0 .
hence proved. 
