Is there a special method to find the the inverse for a matrix which would classified as a lower or left triangular matrix for a matrix L which is n by n. Additionally where the upper part of the matrix would also be all zeros.
where none of the diagonals are equal to zero{(1,1), (2,2),...,(n,n)} ≠ 0 , or a,c,j and j in the example below.
so the determinant ≠ zero.
For example when n=4 finding the inverse of the matrix $L$ where
$$L=\begin{pmatrix}
a & 0 & 0 & 0\\
b & c & 0 & 0\\
d & e & f & 0 \\
g & h & i & j \\
\end{pmatrix}
$$
But could also work when n = 5,6,..., at least 10 (for the sake of simplicity)
Does there exist a method to find the inverse of any sized matrix in this form?
Best Answer
We can write $L = D(I + N)$ where $D$ is diagonal and $N$ is strictly lower triangular and nilpotent ($N^n = 0$): $N_{ij} = L_{ij}/D_{ii}$. Then $L^{-1} = (I+N)^{-1} D^{-1}$.
$D^{-1}$ is diagonal with $(D^{-1})_{ii} = 1/D_{ii}$, and $(I+N)^{-1} = I + \sum_{j=1}^{n-1} (-1)^j N^j$.