= Solution
For each row, a <Gershgorin disc> has centre $a_{jj}=-2$ and radius $r_j=\sum_{k\ne j}|a_{jk}|$. Here $r_1=r_3=1$ and $r_2=2$, so the <Gershgorin circle theorem> gives
$$
D_1=D_3=\{z:|z+2|\le1\},\qquad D_2=\{z:|z+2|\le2\}.
$$
Since $A$ is a real <symmetric matrix>, its <eigenvalues> are real. Intersecting the union of these <Gershgorin discs> with the real axis yields
$$
\boxed{-4\le\lambda\le0.}
$$
This bound permits zero but does not assert a zero <eigenvalue>. The fixed-end <Dirichlet boundary conditions> exclude it: for any nonzero real <vector> $v=(v_1,v_2,v_3)^T$, extend by $v_0=v_4=0$ and calculate
$$
v^T(-A)v=v_1^2+(v_2-v_1)^2+(v_3-v_2)^2+v_3^2
=\sum_{j=0}^{3}(v_{j+1}-v_j)^2>0.
$$
Indeed, equality would force every adjacent value to agree with the zero endpoint. Thus $-A$ is a <symmetric positive-definite matrix>, $A$ is a <negative-definite matrix>, and the <eigenvalues> satisfy \b[$-4\le\lambda<0$]. The same argument works for any number of interior grid points, with prefactor $c^2/h^2>0$ in the sum. The fixed string has no zero-<frequency> rigid translation.
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