Let $A$ be a symmetric, positive definite matrix.
Is is true that the det$ A$(which is the product of eigenvalues) is smaller or equal to the product of diagonal elements of $A$ ?
I could not prove it, and not even sure if the inequality above is true.
Yes. Let $A=LL^\ast$ be a Cholesky decomposition. Then $\det(A)$ is the product of the squared diagonal elements of $L$, while the product of the diagonal elements of $A$ is the product of the squared norms of the rows of $L$. Since the norm of each row of a matrix is necessarily greater than or equal to the magnitude of the diagonal element on that row, the conclusion follows.