Question

If nxn matrices E and F have the property that EF=I, the E and F commute. Explain why.

Answer 1

Since you know EF=I, you know that both matrices E and F are invertible. Therefore, by algebra you can write\[E ^{-1}EF=E^{-1}I\] simplifying that you get the result\[IF=E^{-1}I\]\[F=E^{-1}\] continuing algebra you can multiply both sides by the inverse of F\[FF^{-1}=E^{-1}F^{-1}\] again simplifying you get to the result\[I=E^{-1}F^{-1}\] which from the given equation EF=I you know that F is the inverse of E and that E is the inverse of F so making that substitution you get\[I=FE\] so by setting your initial equation equal to this end result here you get\[EF=FE\] Hope this helped some.