Triangularity of an inverse matrix
= 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.