If we have a lower triangular matrix $$A=\left(\begin{array}{rrrrr}a_{1,1}&0&0&\cdots&0\\a_{1,2}&a_{2,2}&0&\cdots&0\\a_{1,3} &1_{2,3}&a_{3,3}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\a_{1,n}&a_{2,n}&a_{3,n}&\cdots&a_{n,n}\end{array}\right)$$ does there exist a formula for a certain element of $A^{-1}$ in terms of its coordinates, the elements of $A$, and $\det(A)$?
Given such a matrix of any particular size, we can calculate each term of the inverse matrix by expanding the multiplication and solving. For instance, if $A$ is a $3\rm x 3$ matrix, then we have $$A^{-1}=\left(\begin{array}{lll} \frac 1{a_{1,1}}&0&0\\ -a_{1,2}\over a_{1,1}a_{2,2}&\frac 1{a_{2,2}}&0\\ a_{1,2}a_{2,3}-a_{1,3}a_{2,2}\over a_{1,1}a_{2,2}a_{3,3}&-a_{2,3}\over a_{2,2}a_{3,3}&\frac 1{a_{3,3}} \end{array}\right).$$ We can, by similar methods, calculate exact formulae for any specific value of $n$. These formulae seem tantalizingly as if there is some general formula that holds for all $n$ for each entry, but I am unable to find one. Does such a formula exist?
Note: I am aware of a few other questions on this subject, but none of them seem to answer this; however, I know very little linear algebra I may have just failed to understand them.