Let there exists a matrix B such that  $AB{A}^T=N$, where $A=\left[\begin{array}{ccc}1& 2& 0\\ 2& 1& 0\\ 0& 0& 1\end{array}\right]$ and N is a diagonal matrix of form  $N=diag(n_1,n_2,n_3)$ where $n_1,n_2,n_3$ are three values of n satisfying the equation  $|A-nI|=0,n_1<n_2<n_3$.Trace of matrix $A^{20}$ is equal to $3^{20}+k$,then  $k+n_1+n_2+n_3=?$

$$\begin{gathered} |A-nI|=0\left|\begin{array}{l}1-n20\\ 21-n0\\ 001-n\end{array}\right|={(1-n)}^3-4(1-n) \\ \Rightarrow n=-1,1,n_1=-1,n_2=1,n_3=3 \\ \Rightarrow A^3-3A^2+3I=O \\ \left|A\right|\left|B\right|{\left|A\right|}^T=\left|N\right|=n_1{n}_2{n}_3=-3={\left|A\right|}^2\left|B\right| \\ \left|B\right|=\frac{1}{3} \\ A=\left[\begin{array}{l}100\\ 010\\ 001\end{array}\right]+\left[\begin{array}{l}020\\ 200\\ 000\end{array}\right]=I+B \\ {B}^2=4\left|\begin{array}{l}100\\ 010\\ 000\end{array}\right|,B^3=8\left|\begin{array}{l}010\\ 100\\ 000\end{array}\right| \\ Tr\left(A^k\right)=3+2({}^{k}C_2{2}^2+{}^{k}C_4{2}^4+{}^{k}C_6{2}^6+\mathrm{........}) \\ =1+2({}^{k}C_0{}^{k}C_2{2}^2+{}^{k}C_4{2}^4+\mathrm{........})=1+3^k+{(-1)}^k \end{gathered}$$