I see this problem as asking, "given the matrix [tex]E[/tex], find [tex]E^3[/tex]".
We can do this directly by computing the matrix product [tex]E^3=EEE[/tex].
[tex]E=\begin{bmatrix}2a&0&-a\\0&1&2\\-3&a&0\end{bmatrix}[/tex]
[tex]\implies E^2=\begin{bmatrix}2a&0&-a\\0&1&2\\-3&a&0\end{bmatrix}\begin{bmatrix}2a&0&-a\\0&1&2\\-3&a&0\end{bmatrix}=\begin{bmatrix}4a^2+3a&-a^2&-2a^2\\-6&2a+1&2\\-6a&a&5a\end{bmatrix}[/tex]
[tex]\implies E^3=\begin{bmatrix}4a^2+3a&-a^2&-2a^2\\-6&2a+1&2\\-6a&a&5a\end{bmatrix}\begin{bmatrix}2a&0&-a\\0&1&2\\-3&a&0\end{bmatrix}=\begin{bmatrix}8a^3+12a^2&-2a^3-a^2&-4a^3-5a^2\\-12a-6&4a+1&10a+2\\-12a^2-15a&5a^2+a&6a^2+2a\end{bmatrix}[/tex]
