Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $B = (b_{i,j})_{1 \leqslant i,j \leqslant n}$ be the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by
$$b_{1,2p-1} = 1 \quad b_{2p-1,1} = 1 \quad b_{p,p} = -2$$
all other coefficients of $B$ being zero.
Determine the ordered spectrum of matrix $B$.