<s>[INST] Calculate the determinant of $A-2A^{-1}$ given the characteristic and minimal polynomials of $A$ Given the characteristic polynomial and minimal polynomial of $A$ being $(x+2)^{6}(x-1)^{3}$ and $(x+2)^{2}(x-1)^{3}$, respectively. How do I determine the characteristic polynomial and minimal polynomial of $A-2A^{-1}$.
 I tried to algebraically rearrange this in order to have two separate determinant calculations but haven't had success so far. 
Also I have the Jordan Canonical Form of $A$ and must determine that of the $A-2A^{-1}$ matrix. 
 [/INST] $A$ is similar to a triangular matrix with a diagonal of $-2$ repeated $6$ times and $1$ repeated $3$ times. Therefore, $\det A= (-2)^6 \cdot 1^3 = 2^6$. For simplicity, let's assume $A$ is triangular.
We have $\det (A-2A^{-1}I) \det A = \det (A^2-2I)$.
The diagonal of $A^2$ is $(-2)^2$ repeated $6$ times and $1^2$ repeated $3$ times.
Therefore, the diagonal of $A^2-2I$ is $(-2)^2-2$ repeated $6$ times and $1-2$ repeated $3$ times and so $\det  (A^2-2I)=-2^6$.
Therefore, $\det (A-2A^{-1}I)=-1$.
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