# Inverse of upper triangular matrix

I have an upper triangular matrix that I want to solve the inverse for. I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity matrix.

If I'm solving for a specific $x_i$, do I need to work with the whole matrix $A$ or just a certain part of it?

I'm thinking I have to use all of matrix $A$, but because it's already in upper triangular form - hence echelon form, I don't need the entirety of $A$.

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