2015 Putnam A6

Let $n$ be a positive integer. Suppose that $A$, $B$, and $M$ are $n\times n$ matrices with real entries such that $AM = MB$, and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\det(A-MX) = \det(B-XM)$ for every $n\times n$ matrix $X$ with real entries.

令 $n$ 为正整数。假设 $A$、$B$ 和 $M$ 是具有实数项的 $n\times n$ 矩阵，使得 $AM = MB$，并且 $A$ 和 $B$ 具有相同的特征多项式。证明 $\det(A-MX) = \det(B-XM)$ 对于每个具有实数项的 $n\times n$ 矩阵 $X$。