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  4. Let A=[1 0 2 2 0 1 1 1 2], then det((A-I)3-4 A) is

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Q. Let $A=\begin{bmatrix}1 & 0 & 2 \\ 2 & 0 & 1 \\ 1 & 1 & 2\end{bmatrix}$, then ${det}\left((A-I)^{3}-4 A\right)$ is

Matrices

Solution:

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