Check if the determinant of a positive definite matrix equals the product of its diagonal elements, then the matrix is diagonal

Question

Let $A=[a_{ij}]\in M_n, $ and $A$ is a positive definite matrix, Show that $A$ is diagonal if $det(A)= \prod_{i=1}^n a_{ii}$.

I know a positive definite matrix is also Hermitian, so it remains to show nondiagonal entries equals 0. But I don't know how to use the determinant condition to prove this statement. Please give some suggestions? Thank you!