which is obtained by appending the identity matrix on the right of U.
Then using row operations we reduce the left half of this augmented
matrix to the identity matrix, and we are left with the inverse of U in
the right half of the augmented matrix.
Now if we start from the bottom working our way up with the elimination
process the right half of the augmented matrix remains upper triangular, hence
when we finish with the identity in the left half of the augmented matrix
we have an upper triangular matrix in the right half of the augmented
matrix. But this upper triangular is the inverse of U, which proves that
the inverse is upper triangular.