<s>[INST] Is the derivative of the characteristic polynomial equal to the sum of characteristic polynomial of principal submatrices? Let $A$ by an $n \times n$ matrix over the complex numbers and let $\phi(A,x) = \det(xI-A)$ be the characteristic polynomial of $A$.  Let $B_i$ be the principal submatrix of $A$ formed by deleting the $i$-th row and column of $A$. Let $\phi(B_i,x)$ be the characteristic polynomial of $B_i$.
Is it always true that the derivative of $\phi(A,x)$ is the sum of $\phi(B_i,x)$ as $i$ ranges across the rows/columns of $A$?
That is, is the following an identity?
$$ \frac{d\ \phi(A,x)}{dx} = \sum_i \phi(B_i,x) $$
If not, what is a small counterexample?  (The equation works for all $2 \times 2$ matrices and all adjacency matrices of graphs....)
 [/INST] It is true.
Proof:
$$
\phi(A,x) =
\begin{pmatrix}
  x-a_{1,1} & -a_{1,2} & \cdots & -a_{1,n} \\
  -a_{2,1} & x-a_{2,2} & \cdots & -a_{2,n} \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  -a_{m,1} & -a_{m,2} & \cdots & x-a_{m,n}
 \end{pmatrix}
$$Let us write the derivative using multilinearity:
$$
\phi'(A,x) =
\begin{pmatrix}
  1 &0 & \cdots & 0 \\
  -a_{2,1} & x-a_{2,2} & \cdots & -a_{2,n} \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  -a_{m,1} & -a_{m,2} & \cdots & x-a_{m,n}
 \end{pmatrix} +
\begin{pmatrix}
  x-a_{1,1} & -a_{1,2} & \cdots & -a_{1,n} \\
  0 & 1 & \cdots & 0 \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  -a_{m,1} & -a_{m,2} & \cdots & x-a_{m,n}
 \end{pmatrix} +\dots +
\begin{pmatrix}
  x-a_{1,1} & -a_{1,2} & \cdots & -a_{1,n} \\
  -a_{2,1} & x-a_{2,2} & \cdots & -a_{2,n} \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  0 & 0 & \cdots & 1
 \end{pmatrix}
= \sum _{i=1}^n \phi(B_i,x)
$$
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