Finding eigenvalues. I'm working on the following problem:
Define $T \in L(F^n)$ (T an operator) by
$T(x_1,...,x_n) = (x_1+...+x_n,...,x_1+...+x_n)$
Find all eigenvalues and eigenvectors of $T$.
I've found that the eigenvalues of $T$ are $\lambda = 0$ and $\lambda = n$. Is there an easy way to prove that these are the only eigenvalues of $T$? Determining and solving the characteristic polynomial is messy for arbitrary $n$.

$|A|=0$
Product of eigen values $=0$
So atleast one eigen value is $0$
matrix is all one,So its rank is $1$(Echelon form)
As Casteels said,rank of a matrix is the number of non-zero eigenvalue
Also Sum of eigen value $ =n$
So other eigen value is $n$
