Question

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 \(M ?\)

Answer 1

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 \(c\).