Let $\varphi$ and $\psi$ be a linear transformation in vector space $V$, and given the inverse linear transformation of $\varphi$ exists, and the matrix representation of $\varphi$ and $\psi$ on the first basis of $V$ to be matrices $A$ and $B$, respectively, the transition matrix of $V$ from the first basis to second basis is $P$. Find the matrix representation $\psi\varphi^{-2}+2\varphi+I$ (where $I$ is the identity transformation on $V$) on the second basis of $V$.
I should find the matrix representation of $\psi\varphi^{-2}+2\varphi+I$ separately, i.e. find the martix representation of $\psi\varphi^{-2}$, $2\varphi$ and $I$, respectively, by linearity property. What makes me feel difficulty is to find the matrix representation of $\psi\varphi^{-2}$, any idea to deal with it?
[Transition matrix in here refers to the matrix associated with a change of basis for a vector space.(Source: Wikipedia)]
The answer given for this question is $P^{-1}BA^{-2}P+2P^{-1}AP+I_{n}$. Thanks in advance!
Hint:
remember that, if the inverse exist, $(AB)^{-1}=B^{-1}A^{-1}$,
so also:
$(P^{-1}AP)^{-1}=P^{-1}A^{-1}P$
and note that:
$(P^{-1}MP)^{2}=(P^{-1}MP)(P^{-1}MP)=P^{-1}M^2P$