How to find the inverse of an upper/lower triangular matrix directly without any messy calculation just like inverse of diagonal matrix is obtained by inverting all the diagonal entries...

Question

anonymous · Answer

To my knowledge there's no way to just write down the inverse of a triangular matrix like you can for a diagonal one.  If you were use the cofactor method, then most of the cofactors would be zero, so in that sense it would be easier to evaluate than the inverse of an arbitrary matrix but that's about as far as it goes.

mathmate · Answer

To invert a triangular matrix is kind of half way between a full matrix and a diagonal matrix.

The forward pass is already done (or need not be done).  To get the inverse, we just need to augment a square matrix as a bookkeeping matrix and do the back substitution.  The bookkeeping matrix will then be the inverse of the triangular matrix.