Proof of Theorem UMI

By Definition UM, we know that $\adjoint{U}U=I_n$. The matrix $I_n$ is nonsingular (since it row-reduces easily to $I_n$, Theorem NMRRI). So by Theorem NPNT, $U$ and $\adjoint{U}$ are both nonsingular matrices.

The equation $\adjoint{U}U=I_n$ gets us halfway to an inverse of $U$, and Theorem OSIS tells us that then $U\adjoint{U}=I_n$ also. So $U$ and $\adjoint{U}$ are inverses of each other (Definition MI).