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Homework Statement
Consider an n x n matrix A with the property that the row sums all equal the same number S. Show that S is an eigenvalue of A. [Hint: Find an eigenvector.]
Homework Equations
##Ax=λx##
The Attempt at a Solution
S is just lambda here, so I begin solving this just like you would normally.
##Ax=Sx##
##Ax-Sx = 0##
##(A-SI)x = 0##
Subtracting gives me the matrix: ##\begin{bmatrix}
a_{11}-S & a_{12} & a_{13} \\
a_{21} & a_{22}-S & a_{23} \\
a_{31} & a_{32} & a_{33}-S
\end{bmatrix}##
My problem is that I don't know how to find an eigenvector from this matrix. I can't row reduce because I don't know any of the values of the matrix, and I can't recall any other way to find the eigenvector.