Answer
$$A^{-1}=BC^{-1}$$
$$A^{-1}=U^{-1}L^{-1}P$$
Work Step by Step
$$AB=C$$
Our goal is to find a formula for $A^{-1}$. Note that matrices are not mutually commutative, i.e. $a\cdot{b}\neq{b}\cdot{a}$. Thus, when multiplying both sides of the equation by any matrix, we must ensure that the inverse is always placed on the same side.
Introduce $A^{-1}$ into the equation as a multiple on both sides.
$$A^{-1}AB=A^{-1}C$$
Since $|A^{-1}\cdot{A}|=1$,
$$B=A^{-1}C$$
We proceed to eliminate C from the RHS by multiplying C by its inverse, leaving $A^{-1}$.
$$BC^{-1}=A^{-1}CC^{-1}=A^{-1}$$
We now aim to achieve this same objective with the equation
$$PA=LU$$
As with earlier, introduce $A^{-1}$ to the equation as a multiple on both sides
$$PAA^{-1}=LUA^{-1}$$
$$P=LUA^{-1}$$
$LU$ refers to the product $L\cdot{U}$. To isolate $A^{-1}$, we multiply both sides of the equation by the inverse of this product, $(LU)^{-1}$.
$$(LU)^{-1}P=(LU)^{-1}LUA^{-1}=A^{-1}$$
Since $(LU)^{-1}=U^{-1}L^{-1}$,
$$U^{-1}L^{-1}P=A^{-1}$$
