I have a matrix A and a lower triangular matrix L (with 1's along the diagonal) and an upper triangular matrix U. These are constructed such that $A=LU$. I know that $A^{-1} = L^{-1}U^{-1}$ and I know that the inverse of L is simply the non-diagonal entries with their signs flipped.
Question: Is there an easy way to find the inverse of U?
example: $$\begin{bmatrix}8 & 1 &6\\3 & 5 & 7\\4&9&2\end{bmatrix}^{-1} = \begin{bmatrix}1 & 0 &0\\-.5 & 1 & 0\\-.375 & -.544 & 1\end{bmatrix}\begin{bmatrix}8 & 1 &6\\0 & 8.5 & -1\\0&0&5.294\end{bmatrix}^{-1}$$
I need to find an algorithm for computing the inverse of the far right upper triangular matrix.