$A$ is not merely tridiagonal, it is also Toeplitz. Hence, the $n$ real eigenvalues of $A$ are given by [0]

$$\lambda_k (A) = 2 + 2 \cos \left(\frac{k \pi}{n+1}\right)$$

for $k \in \{1,2,\dots,n\}$. Thus,

$$0 < 2 - 2 \cos \left(\frac{\pi}{n+1}\right) \leq \lambda_k (A) \leq 2 + 2 \cos \left(\frac{\pi}{n+1}\right) < 4$$

and we conclude that $A$ is invertible.

[0] Silvia Noschese, Lionello Pasquini, and Lothar Reichel, Tridiagonal Toeplitz Matrices: Properties and Novel Applications, 2006.

$\mathrm A$ is not merely tridiagonal, it is also Toeplitz. Hence, the $n$ real eigenvalues of $\mathrm A$ are given by [0]

$$\lambda_k (\mathrm A) = 2 + 2 \cos \left(\frac{k \pi}{n+1}\right)$$

for $k \in \{1,2,\dots,n\}$. Thus,

$$0 < 2 - 2 \cos \left(\frac{\pi}{n+1}\right) \leq \lambda_k (\mathrm A) \leq 2 + 2 \cos \left(\frac{\pi}{n+1}\right) < 4$$

and we conclude that $\mathrm A$ is invertible.

[0] Silvia Noschese, Lionello Pasquini, and Lothar Reichel, Tridiagonal Toeplitz Matrices: Properties and Novel Applications, 2006.