How to find the value of $\alpha$

Question

Let $M$ be a $3 \times 3 $ matrix such that

$M \begin{bmatrix} -2 \\ 1\\ 0\end{bmatrix}$ = $\begin{bmatrix} 6 \\ -3\\ 0\end{bmatrix}$ and suppose that

$M^3 \begin{bmatrix} 1 \\ -1/2\\ 0\end{bmatrix}$ = $\begin{bmatrix} \alpha \\ \beta\\ \gamma\end{bmatrix}$ for some $\alpha, \beta,\gamma \in R$ then

$\left|\alpha\right|$ is equal to :

I have figured out that $\begin{bmatrix} -2 \\ 1\\ 0\end{bmatrix}$ is an eigen vector of given matrix with eigen value equal to $-3$ .

But I am stuck after that step. Can somebody help

Thank you.

Answer 1

Hint: If $v$ is an eigenvector of a matrix $A$ with eigenvalue $\lambda$, the it is also an eigenvector of $A^3$ with eigenvalue $\lambda^3$.