TITLE: Generalisation of an expression.
QUESTION [5 upvotes]: I have been solving matrices questions lately and this pattern keeps showing up:

Given matrix $ P = \begin{bmatrix}
1 & 0 & 0\\
4 & 1 & 0\\
16 & 4 & 1\\
\end{bmatrix}$ and $I$ be identity matrix of order 3. If $[Q] = [q_{ij}]$ is a matrix such that $P^{50}-Q= I$ then $\frac{q_{31}+q_{32}}{q_{21}} = ?$

While solving I found this pattern for a general matrix of the type:
$$L = \begin{bmatrix}
1 & 0 & 0\\
a & 1 & 0\\
b & a & 1\\ 
\end{bmatrix}$$
As
$$L^n = \begin{bmatrix}
1 & 0 & 0\\
a×n & 1 & 0\\
K & a×n & 1\\
\end{bmatrix}$$
Where $K = (L_{31})_{{n-1}}+a^2×(n-1) + b$
If I have to figure out $L_{50}$ using this algorithm, I need to find $L_{49}, L_{48}$ and so on which basically defeats the purpose. How do you reiterate the value of $L_{n-1}$ so that I can use directly the value of $b$ to calculate $L_{31}$ of any power of matrix?

REPLY [2 votes]: Note that $L=I+N$ with $N^3=0$. Therefore, $L^n = I+n N + \frac{n(n-1)}{2}N^2$. Since
$$
N^2 = \begin{bmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
a^2 & 0 & 0\\ 
\end{bmatrix}
$$
we get
$$
L^n = \begin{bmatrix}
1 & 0 & 0\\
n a & 1 & 0\\
nb +\frac{n(n-1)}{2}a^2& n a & 1\\ 
\end{bmatrix}
$$