Decompose this matrix as a sum of unit and nilpotent matrix. Show that the matrix $A=\begin{bmatrix} 
        1 & 0  \\
        2 & 1  \\
        \end{bmatrix}$ can be decomposed as a sum of a unit and nilpotent matrix. 
Hence evaluate the matrix $A^{2007}$.
I read about nilpotent matrices the other day. It said "A square matrix such that is the zero matrix for some positive integer matrix power" is nilpotent.
But how to use this in this problem is beyond me.

$A=\begin{bmatrix} 
        1 & 0  \\
        2 & 1  \\
        \end{bmatrix}=\begin{bmatrix} 
        1 & 0  \\
        0 & 1  \\
        \end{bmatrix}+\begin{bmatrix} 
        0 & 0  \\
        2 & 0  \\
        \end{bmatrix}=E+A_1$
$A_1=\begin{bmatrix} 
        0 & 0  \\
        2 & 0  \\
        \end{bmatrix}$ is nilpotent matrix because $A_1^2=\begin{bmatrix} 
        0 & 0  \\
        2 & 0  \\
        \end{bmatrix} \cdot\begin{bmatrix} 
        0 & 0  \\
        2 & 0  \\
        \end{bmatrix}=\begin{bmatrix} 
        0 & 0  \\
        0 & 0  \\
        \end{bmatrix}$
And what about the second question. Take into account that $$A^2=\begin{bmatrix} 
        1 & 0  \\
        2 & 1  \\
        \end{bmatrix}\cdot \begin{bmatrix} 
        1 & 0  \\
        2 & 1  \\
        \end{bmatrix}=\begin{bmatrix} 
        1 & 0  \\
        4 & 1  \\
        \end{bmatrix}$$
$$A^3=\begin{bmatrix} 
        1 & 0  \\
        4 & 1  \\
        \end{bmatrix}\cdot \begin{bmatrix} 
        1 & 0  \\
        2 & 1  \\
        \end{bmatrix}=\begin{bmatrix} 
        1 & 0  \\
        6 & 1  \\
        \end{bmatrix}$$
$$\ldots$$
$$A^n=\begin{bmatrix} 
        1 & 0  \\
        2n & 1  \\
        \end{bmatrix}$$
