Trace of the matrix power Say I have matrix $A = \begin{bmatrix}
a &amp; 0 &amp; -c\\
0 &amp; b &amp; 0\\
-c &amp; 0 &amp; a 
\end{bmatrix}$.

What is matrix trace
tr(A^200)

Thanks much!
For this kind of problems, first you have to get eigenvalues and their corresponding eigenvectors. For this problem,
$$ \lambda_1=b,\lambda_2=a-c,\lambda_3=a+c, v_1=(0,1,0)^T,v_2=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})^T,v_3=(-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})^T. $$
Let 
$$ P=(v_1,v_2,v_3)=\left(\begin{matrix}0&amp;\frac{1}{\sqrt{2}}&amp;-\frac{1}{\sqrt{2}}\\
1&amp;0&amp;0\\
-\frac{1}{\sqrt{2}}&amp;0&amp;\frac{1}{\sqrt{2}}
\end{matrix}\right). $$
Then it is easy to check $P^T=P^{-1},P^TAP=\text{diag}(\lambda_1,\lambda_2,\lambda_3)$. Hence
\begin{eqnarray*}
\text{Tr}(A^{200})&amp;=&amp;\text{Tr}(P^TA^{200}P)\\
&amp;=&amp;\text{Tr}((P^TAP)^{200})\\
&amp;=&amp;\lambda_1^{200}+\lambda_2^{200}+\lambda_3^{200}\\
&amp;=&amp;b^{200}+(a-c)^{200}+(a+c)^{200}.
\end{eqnarray*}
You just need to calculate Eigen values bcoz sum of eigen values =$tr(A)$ and if "a" is the eigen value of matrix $A$ then "$a^n$" is the eigen value of $A^n$.
thus,
$$Tr(A^{200})=b^{200}+(a−c)^{200}+(a+c)^{200}$$