TITLE: Inverse of special upper triangular matrix
QUESTION [9 upvotes]: Consider the following $n \times n$ upper triangular matrix with a particularly nice structure:
\begin{equation}\mathbf{P} = \begin{pmatrix}
1 & \beta & \alpha+\beta & \dots & (n-3)\alpha + \beta & (n-2)\alpha + \beta\\
0 & 1 & \beta & \dots & (n-4)\alpha + \beta & (n-3)\alpha + \beta\\
0 & 0 &     1 & \dots & (n-5)\alpha + \beta & (n-4)\alpha + \beta\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \dots & \beta & \alpha+\beta\\
0 & 0 & 0 & \dots & 1 & \beta\\
0 & 0 & 0 & \dots & 0 & 1
\end{pmatrix}
\end{equation}
i.e. 
\begin{equation}
p_{i,j}=\begin{cases}
0, &i>j,\\
1, &i=j,\\
(j-i-1)\alpha+\beta, &i<j.
\end{cases}
\end{equation}
Would it be possible to find an explicit expression for the elements of the inverse of $\mathbf{P}$?

REPLY [3 votes]: Working out the the details of the answer given by user WhatsUp, the elements of the inverse of $\mathbf{P}$ are given by
\begin{equation}
p^{(-1)}_{i,j} =\begin{cases}
0, &i>j,\\
1, &i=j,\\
\frac{\lambda^2(1+\lambda)^{j-i-1}-\mu^2(1+\mu)^{j-i-1}}{\lambda-\mu},&i<j.
\end{cases}
\end{equation}