What's wrong in my example about Q-matrix?

Question

I find the next property about Q-matrices: $$ \text{Let Q a Q-Matrix, then} \frac{d^k P(0)}{dt^k}=Q^k ; \text{ for } k=0,1,2,\cdots.$$ I was trying to verify the property with the next example: $Q=\begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix}$ then $$P(s)=\begin{pmatrix} e^{-s} & e^s \\ e^s & e^{-s} \end{pmatrix} $$ $$\frac{d^k P(0)}{ds^k}=\begin{pmatrix} (-1)^ke^{-0} & e^0 \\ e^0 & (-1)^ke^{-0} \end{pmatrix}=\begin{pmatrix} (-1)^k & 1 \\ 1 & (-1)^k \end{pmatrix}\neq Q^k$$ But it contradicts the proposition!! I want to know what is wrong in my example or interpretation.

Answer 1

If $P(s)$ and $Q$ are as above, then we have

$\frac{d^k P(0)}{ds^k}=Q^k$ only for $k=1,3,5,...$.