So my problem is this i have a matrix $4\times4$ $G$ which is unknown except for its determinant which is $2^{11}$ and i also have this expression $G^2-36G + 27I = 0$
And what im asked is to calculate the determinant of this $4^{-1}G-9I$ and I know it's $4^{-4}\det(G-9I)$, but the problem is that i got stuck because of that $-$ symbol
So if someone can help me solve this I would appreciate it a lot
You forgot to factor out the 1/4 from the whole expression.
$\text{det}(4^{-1}G-9I)=2^{-8}\text{det}(G-36I)$ was what you were looking for.
From there, notice that $G^2-36G+27I=G(G-36I)-27I$
so since the left hand side is $0$, we find that
$G(G-36I)=27I$ and
$\text{det}(G)\text{det}(G-36I)=27^4\text{det}(G-I)$ by multiplicativity of the determinant
And so it follows that $\text{det}(4^{-1}G-9I)=2^{-19}*3^{12}$