<s>[INST] On the eigenvalue of a particular kind of matrix 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 
 [/INST] 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
$$
\left(
\begin{array}{c}
x^4 \\
x^3 y \\
x^2 y^2 \\
x y^3 \\
y^4
\end{array}
\right)
$$
</s>