4. Let $C_m \in \mathbb{R}[Y]$ be the characteristic polynomial of $M_m$. a. Verify that $C_1 = Y$, $C_2 = Y^2 - 2$ and $$C_m = Y C_{m-1} - m(m-1) C_{m-2}, \quad m \geq 3$$ b. Calculate the determinant of $M_m$. c. Prove that, if $e_m$ denotes the integer part of $m/2$, $$C_m = \sum_{k=0}^{e_m} (-1)^k c_{m,k} Y^{m-2k}$$ with $$c_{m,0} = 1; \quad c_{m,k} = \sum_{(a_1, \ldots, a_k) \in J_k(m)} a_1(a_1+1) a_2(a_2+1) \cdots a_k(a_k+1), \quad 1 \leq k \leq e_m;$$ where $J_k(m)$ denotes the set of $k$-tuples of integers from $\{1, \ldots, m-1\}$ such that $a_i + 2 \leq a_{i+1}$ for $1 \leq i \leq k-1$.
4. Let $C_m \in \mathbb{R}[Y]$ be the characteristic polynomial of $M_m$.\\
a. Verify that $C_1 = Y$, $C_2 = Y^2 - 2$ and
$$C_m = Y C_{m-1} - m(m-1) C_{m-2}, \quad m \geq 3$$
b. Calculate the determinant of $M_m$.\\
c. Prove that, if $e_m$ denotes the integer part of $m/2$,
$$C_m = \sum_{k=0}^{e_m} (-1)^k c_{m,k} Y^{m-2k}$$
with
$$c_{m,0} = 1; \quad c_{m,k} = \sum_{(a_1, \ldots, a_k) \in J_k(m)} a_1(a_1+1) a_2(a_2+1) \cdots a_k(a_k+1), \quad 1 \leq k \leq e_m;$$
where $J_k(m)$ denotes the set of $k$-tuples of integers from $\{1, \ldots, m-1\}$ such that $a_i + 2 \leq a_{i+1}$ for $1 \leq i \leq k-1$.