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Let $M$ be a $2 \times 2$ matrix with the property that the sum of each of the rows and also the sum of each of the columns is the same constant $c$. Which (if any) any of the vectors
$U:=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \quad V:=\left(\begin{array}{l} 0 \\ 1 \end{array}\right), \quad W:=\left(\begin{array}{l} 1 \\ 1 \end{array}\right),$
must be an eigenvector of $M ?$
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Let $M=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$, then the sum of each row and each column is equal to $c$, so we have the equations

\begin{aligned} & a+b =c \\ & c+d =c \\ & a+c =c \\ & b+d =c \\ \end{aligned}
Solving these equations, we find that $a=b=c=d$. Hence, the matrix $M$ is a scalar matrix with all entries equal to $c$. It follows that both $U$ and $V$ are eigenvectors of $M$ with eigenvalue $C$, and $W$ is also an eigenvector with eigenvalue $c$.
ago by Platinum (164,236 points)

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