Calculating the Inverse of a matrix. $A)$
Solve:
$x_1-x_2=1$
$-x_1+2x_2-x_3=0$
$\vdots$
$-x_{99}+2x_{100}=0$
$B$)Deduce the inverse of 
A=\begin{pmatrix}
        1 & -1 & 0 & \cdots& \cdots & 0   \\
        -1 & 2 & -1 & \ddots& \cdots  &\vdots\\
        0 & -1 & 2 &-1& \ddots& \vdots\\
        \vdots & \ddots & \ddots & \ddots &\ddots&0\\
        \vdots & \cdots & \ddots & \ddots &2&-1\\
        0 & \cdots & \cdots & 0 &-1&2\\
\end{pmatrix}
I've solved the first part and got:
$x_1=100$
$x_2=99$
$\vdots$
$x_{100}=1$ (Correct me if I'm wrong)
For the part B , I'm not sure how to approach it although I'm quite sure the first column of $A^{-1}$ should contain $x_1$ till $x_{100}$ in this order , however I have no idea on how to fill the other columns.
If anyone could help me or give me hints , I would be grateful.
Thanks in advance.

$\begin{pmatrix} 
1 & -1 \\
-1 & 2 
\end{pmatrix}^{-1}=\begin{pmatrix} 
2 & 1 \\
1 & 1 
\end{pmatrix}$
$\begin{pmatrix} 
1 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 2
\end{pmatrix}^{-1}=\begin{pmatrix} 
3 & 2 & 1\\
2 & 2 & 1\\
1 & 1 & 1
\end{pmatrix}$
$\begin{pmatrix} 
1 & -1 & 0 & 0\\
-1 & 2 & -1 & 0\\
0 & -1 & 2 & -1\\
0 & 0 & -1 & 2
\end{pmatrix}^{-1}=\begin{pmatrix} 
4 & 3 & 2 & 1\\
3 & 3 & 2 & 1\\
2 & 2 & 2 & 1\\
1 & 1 & 1 & 1
\end{pmatrix}$
Can you guess the inverse of $A$?
