Putnam 1984 A3

Problem A-3 Let $n$ be a positive integer. Let $a, b, x$ be real numbers, with $a \neq b$, and let $M_{n}$ denote the $2 n \times 2 n$ matrix whose ( $i, j$ ) entry $m_{i j}$ is given by $$m_{i j}=\left\{\begin{array}{ll} x & \text { if } i=j, \\ a & \text { if } i \neq j \\ b & \text { if } i \neq j \end{array} \text { and } i+j \text { is even }, ~ i+j\right. \text { is odd. }$$

Thus, for example, $M_{2}=\left(\begin{array}{llll}x & b & a & b \\ b & x & b & a \\ a & b & x & b \\ b & a & b & x\end{array}\right)$. Express $\lim _{x \rightarrow a} \operatorname{det} M_{n} /(x-a)^{2 n-2}$ as a polynomial in $a, b$, and $n$, where $\operatorname{det} M_{n}$ denotes the determinant of $M_{n}$.