Let $x+y=1$. Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&x^5+y^5&5x^5&10x^5\\10y^5&10y^5&5y^5&x^5+y^5&5x^5\\5y^5&10y^5&10y^5&5y^5&x^5+y^5 \end{pmatrix} $
Is $1$ an eigenvalue of $A$ ?
It is not obvious to me if $1$ is an eigenvalue or not (the row sums of $A$ are not all equal ... if they were equal then $1$ would obviously be an eigenvalue ).
Please help
If $x = \frac{g}{g+h}$ and $y = \frac{h}{g+h},$ then $$ \left( \begin{array}{c} g^4 \\ g^3 h \\ g^2 h^2 \\ g h^3 \\ h^4 \end{array} \right) $$ is an eigenvector with eigenvalue $1.$ So is the multiple $$ \left( \begin{array}{c} x^4 \\ x^3 y \\ x^2 y^2 \\ x y^3 \\ y^4 \end{array} \right) $$