Triangularity of an inverse matrix

An invertible lower triangular matrix has a lower triangular matrix inverse. To see this, solve $Ax=e_j$ for column $j$ of the inverse. The nonzero diagonal and forward substitution force $x_i=0$ for $i<j$. Applying the result to the transpose proves the upper-triangular case as well. Invertibility is essential; a singular triangular matrix has no ordinary matrix inverse.