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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.

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    $\begingroup$ Do you want absolute values? Otherwise $\begin{bmatrix}1&2&0\\2&1&0\\0&0&1\end{bmatrix}$ would be a counterexample. Do you want to assume invertibility? Otherwise the matrix with all $1$s is a counterexample. $\endgroup$ Jul 26, 2014 at 17:01
  • $\begingroup$ Thank you very much for your examples! I don't want absolute values or invertible matrices. $\endgroup$ Jul 26, 2014 at 17:16

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

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