Question

Let $\beta$ be a real number. Consider the matrix
$A=\begin{pmatrix}\beta & 0 & 1 \\\2 & 1 & -2 \\\3 & 1 & -2\end{pmatrix}$
If $A^7-(\beta-1) A^6-\beta A^5$ is a singular matrix, then the value of $9 \beta$ is __.

Answer 1

The answer is 3.