Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right]$. Then the sum of the diagonal elements of the matrix ${\left(A+I\right)}^{11}$ is equal to:

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations

$\lambda x+y+z=1$

$x+\lambda y+z=1$

$x+y+\lambda z=1$
is inconsistent, then $\sum _{\lambda \in S}\left({\left|\lambda \right|}^2+\left|\lambda \right|\right)$ is equal to