Question

Given the following matrix, where a is constant, find the indicated matrix:

Answer 1

I see this problem as asking, "given the matrix $E$, find $E^3$".

We can do this directly by computing the matrix product $E^3=EEE$.

$E=\begin{bmatrix}2a&0&-a\\0&1&2\\-3&a&0\end{bmatrix}$

$\implies E^2=\begin{bmatrix}2a&0&-a\\0&1&2\\-3&a&0\end{bmatrix}\begin{bmatrix}2a&0&-a\\0&1&2\\-3&a&0\end{bmatrix}=\begin{bmatrix}4a^2+3a&-a^2&-2a^2\\-6&2a+1&2\\-6a&a&5a\end{bmatrix}$

$\implies E^3=\begin{bmatrix}4a^2+3a&-a^2&-2a^2\\-6&2a+1&2\\-6a&a&5a\end{bmatrix}\begin{bmatrix}2a&0&-a\\0&1&2\\-3&a&0\end{bmatrix}=\begin{bmatrix}8a^3+12a^2&-2a^3-a^2&-4a^3-5a^2\\-12a-6&4a+1&10a+2\\-12a^2-15a&5a^2+a&6a^2+2a\end{bmatrix}$