Estimates for the Eigenvalues of Matrices

For matrices whose eigenvalues are real (such as Hermitian or real symmetric matrices), we derive simple explicit estimates for the maximal $(\lambda_{\max})$ and the minimal $(\lambda_{\min})$ eigenvalues in terms of determinants of order less than 3. For $3\times3$ matrices, we derive sharper estimates, which use $\det A$ but do not require to solve cubic equations.