Question

Let $A=\left[\begin{array}{ccc}\alpha & -2& -1\\ 4& \beta -1& 4\\ 3& 3& \gamma -1\end{array}\right],$
where $\alpha$ is remainder when $3^{100}$ is divided by $10$ and $\beta ,\gamma$ are the roots of the equation $x^2-9x+20=0(\gamma >\beta ).$
Then the value of $|adj\left(adjA\right)|$ is

Answer 1

$3^{100}=9^{50}=(10-1{)}^{50}=1+10k$
$\therefore \alpha =1$
The roots of equation $x^2-9x+20=0(\gamma >\beta )$
$\Rightarrow \beta =4,\gamma =5$
$\Rightarrow A=\left[\begin{array}{ccc}1& -2& -1\\ 4& 3& 4\\ 3& 3& 4\end{array}\right]$
$|A|=5|adj\left(adjA\right)|=|A{|}^{(n-1{)}^2}=5^4=625$