I want to know whether the $\mbox{det}(A)\neq0$, $$A = \left( {\begin{array}{*{20}{c}} {b_1^2 + {e_1}}&{{b_1}{b_2}{\rho _{12}}}&{{b_1}{b_3}{\rho _{13}}}\\ {{b_1}{b_2}{\rho _{12}}}&{b_2^2 + {e_2}}&{{b_2}{b_3}{\rho _{23}}}\\ {{b_1}{b_3}{\rho _{13}}}&{{b_2}{b_3}{\rho _{23}}}&{b_3^2 + {e_3}} \end{array}} \right)$$ where $b_i,e_i>0$ for $i=1,2,3$, and $\rho_{ij}\in (-1,1)$.
From my tests with some values for the variables, det(A) is not zero. But I do not know whether it holds. Appreciate any suggestion!
One simple sufficient condition for $\det(A) > 0$ is the coefficients $\rho_{ij}$ satisfy an extra inequality: $$1 - \rho_{12}^2 - \rho_{23}^2 - \rho_{13}^2 + 2\rho_{12}\rho_{23}\rho_{13} \ge 0\tag{*1}$$ If this is satisfied, the part of matrix excluding the $e_k$ pieces will be positive semi-definite. Together with the $e_k$ pieces, the matrix become positive definite and hence $\det(A) > 0$. The condition on $\rho_{ij}$ can be rewritten as $$| \rho_{13} - \rho_{12}\rho_{23} | \le \sqrt{1 - \rho_{12}^2} \sqrt{1 - \rho_{23}^2}$$ If $\rho_{ij}$ are correlations among random variables $X_i$ and $X_j$, above condition can be interpreted as:
To construct a counter-example for the problem, one start from a configuration where $(*1)$ is violated the most. For example, I start with the configuration
$$\rho_{12} = \rho_{23} = 1,\quad\rho_{13} = 0$$ perturb it a little bit to get
$$\begin{cases} e_1 = e_2 = e_3 = t,\\ b_1 = b_2 = b_3 = 1,\\ \rho_{12} = \rho_{23} = 1-t,\\ \rho_{13} = 0 \end{cases} \quad\iff\quad A(t) = \begin{bmatrix} 1 + t & 1-t & 0\\ 1 - t & 1+t & 1-t\\ 0 & 1 -t & 1 + t \end{bmatrix} $$ It is easy to check $$\det(A(t)) = -(t+1)(t^2 - 6t + 1) \quad\text{ and }\quad \det(A(0)) < 0 $$ Solving the quadratic equation $t^2 - 6t + 1 = 0$ gives us $t = 3 - \sqrt{8}$. For this particular $t$, the corresponding $A(t)$ has the form specified in question and yet $\det(A(t)) = 0$.