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

Select the correct choice below.

A. According to the Invertible Matrix Theorem, E and F both have columns that span R". So FE = EF. Thus, E and F commute.

B. According to the Invertible Matrix Theorem, E and F must be invertible and inverses. So FE =I and I EF. Thus, E and F commute.

C. According to the Invertible Matrix Theorem, E and F must not be invisible and therefore cannot be inverses so FE 1 and = EF Thus E and F commute O

D. According to the Invertible Matrix Theorem, E and F must both be identity matrices. So F = I and E = 1; therefore, FE = EF. Thus, E and F commute.

Answer

The correct choice from the provided context is:

B. According to the Invertible Matrix Theorem, E and F must be invertible and inverses. So FE = I and I = EF. Thus, E and F commute.

Explanation:

If ( EF = I ), then multiplying both sides by ( F ) gives ( FE = I ).

This thus leads to ( FE = EF ), showing that ( E ) and ( F ) commute.

The other options are incorrect because:

Thus, B is the correct explanation of why ( E ) and ( F ) commute.