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We explore a certain family $\{A_n\}_{n=1}^{\infty}$ of $n \times n$tridiagonal real symmetric matrices. After deriving a three-term recurrencerelation for the characteristic polynomials of this family, we find a closedform solution. The coefficients of these characteristic polynomials turn out toinvolve the diagonal entries of Pascal's triangle in a tantalizingly predictivemanner. Lastly, we explore a relation between the eigenvalues of variousmembers of the family. More specifically, we give a sufficient condition on thevalues $m,n \in \mathbb{N}$ for when $\texttt{spec}(A_m)$ is contained in$\texttt{spec}(A_n)$. We end the paper with a number of open questions, one ofwhich intertwines our characteristic polynomials with the Fibonacci sequence inan intriguing manner involving ellipses.

Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence