Question

Let $A=\left[\begin{array}{ccc}1& 0& 2\\ 2& 0& 1\\ 1& 1& 2\end{array}\right]$, then $det\left((A-I{)}^3-4A\right)$ is

Answer 1

Answer: 8

Characteristic equation of $A$ is $|A-xI|=0$
$\Rightarrow x^3-3x^2-x-3=0$
or $A^3-3A^2-A-3I=0$
$(A-I{)}^3=4A+2I$
$\left((A-I{)}^3-4A\right)=2I$