Answer
PROOF: If $n \times n$ matrices $E$ and $F$ have the property that $EF=I$, then $E$ and $F$ commute.
Work Step by Step
For some $\vec{x}$ in $\mathbb{R}^n$, let:
$$\vec{b}=F\vec{x}$$
Because $F$ is a $n\times n$ matrix. The product $\vec{b}$ is also in $\mathbb{R}^n$.
$$E(F\vec{x})=E\vec{b}$$
Also, according to associative law of matrix multiplication:
$$E(F\vec{x})=(EF)\vec{x}=I\vec{x}=\vec{x}$$
In conclusion:
$$\vec{x} \mapsto F\vec{x}=\vec{b}$$
$$\vec{b} \mapsto E\vec{b}=\vec{x}$$
E F commute.
