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Notation: $A^{\leftarrow}$ denotes $A$ with the columns reversed; $A^{\uparrow}$ denotes $A$ with the rows reversed; $A^{\leftarrow \uparrow} = A^{\uparrow \leftarrow}$ is denoted $A^{\circ}$ and is the rotation of $A$ by $180^{\circ}$.

Consider first a $(2n+1)\times(2n+1)$ block matrix $\begin{pmatrix} A & v & B^{\circ} \\\\ h & c & h^{\leftarrow} \\\\ B & v^{\uparrow} & A^{\circ} \end{pmatrix}$ where $A, B$ are $n \times n$, $v$ is $n \times 1$, $h$ is $1 \times n$, and $c$ is $1 \times 1$. We have \begin{eqnarray*}\det \begin{pmatrix} A & v & B^{\circ} \\\\ w & c & w^{\leftarrow} \\\\ B & v^{\uparrow} & A^\circ \end{pmatrix}
&=& (-1)^{n(n+1)/2} \det \begin{pmatrix} A & v & B^{\circ} \\\\ B^{\uparrow} & v & A^{\leftarrow} \\\\ w & c & w^{\leftarrow} \end{pmatrix} \tag{permuting rows} \\\\
&=& \det \begin{pmatrix} A & B^{\uparrow} & v \\\\ B^{\uparrow} & A & v \\\\ w & w & c \end{pmatrix} \tag{permuting cols} \\\\
&=& \det \begin{pmatrix} A - B^{\uparrow} & B^{\uparrow}-A & 0 \\\\ B^{\uparrow} & A & v \\\\ w & w & c \end{pmatrix} \tag{subtracting rows} \\\\
&=& \det \begin{pmatrix} A - B^{\uparrow} & 0 & 0 \\\\ B^{\uparrow} & A + B^{\uparrow} & v \\\\ w & 2w & c \end{pmatrix} \tag{adding cols} \\\\
&=& \det (A - B^{\uparrow}) \det \begin{pmatrix} A + B^{\uparrow} & v \\\\ 2w & c \end{pmatrix} \tag{by Leibniz formula}
\end{eqnarray*}

Note that the characteristic polynomial is a special case, since $\chi_M(x) = \det (M - xI)$.

The $(2n)\times(2n)$ block matrix $\begin{pmatrix} A & B^{\circ} \\\\ B & A^{\circ} \end{pmatrix}$ has an almost identical derivation yielding $$\det \begin{pmatrix} A & B^{\circ} \\\\ B & A^{\circ} \end{pmatrix} = \det (A - B^{\uparrow}) \det (A + B^{\uparrow})$$