Relation on the determinant of a matrix and the product of its diagonal entries?

Question

Let $A$ be a $3\times 3$ symmetric matrix, with three real eigenvalues $\lambda_1,\lambda_2,\lambda_3$, and diagonal entries $a_1,a_2,a_3$, is it true that \begin{equation*} \det A=\lambda_1\lambda_2\lambda_3\geq a_1a_2a_3\ ? \end{equation*}

Thank you very much.

Answer 1

Here's a positive definite counterexample: $$\begin{bmatrix}2&1&0\\1&2&0\\0&0&1\end{bmatrix}.$$