  Given a block triangular matrix $M$ over a noncommutative ring with
invertible diagonal blocks, this work gives two new representations of its
inverse $M^{-1}$. Each block element of $M^{-1}$ is explicitly expressed via a
quasideterminant of a submatrix of $M$ with the block Hessenberg type.
Accordingly another representation for each inverse block is attained, which is
in terms of recurrence relationship with multiple terms among blocks of
$M^{-1}$. The latter result allows us to perform an off-diagonal rectangular
perturbation analysis for the inverse calculation of $M$. An example is given
to illustrate the effectiveness of our results.
