Let $A=\left[\begin{array}{cc}3& 1\\ -16& -5\end{array}\right]$, then the sum of all the elements of $A^{50}$ is equal to

Let $t$ be the trace of the matrix 

$A=\left[\begin{array}{ccc}\frac{\left|x+y\right|}{\left|x\right|+\left|y\right|}& {\alpha }_1& {\beta }_1\\ {\alpha }_2& \frac{\left|y+z\right|}{\left|y\right|+\left|z\right|}& {\beta }_2\\ {\alpha }_3& {\beta }_3& \frac{\left|z+x\right|}{\left|z\right|+\left|x\right|}\end{array}\right],$ then