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This question already has an answer here:

Let $a$ is a constant. For which condition(s) this matrix has an inverse?

$$\left[ \begin{array}{cccccc} a & 1 & 0 & \cdots & & 0 \\ 1 & a & 1 & \cdots & & 0 \\ 0 & 1 & \ddots & & & \\ \vdots & \vdots & & a & 1 & 0 \\ & & & 1 & a & 1 \\ 0 & 0 & \cdots & 0 & 1 & a \\ \end{array} \right]$$

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marked as duplicate by Dietrich Burde, Community Jun 16 '15 at 12:17

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    $\begingroup$ Any thoughts of your own? Did you try solving it for the $2\times 2$ and $3\times 3$ cases? $\endgroup$ – Casteels Jun 16 '15 at 12:09
  • $\begingroup$ the eigenvalues of the tridiagonal matrix with all ones on the sub and super diagonals are well known. $\endgroup$ – abel Jun 16 '15 at 12:13