Eigenvalues of a matrix with only one non-zero row and non-zero column. Here is the full question. 


*

*Only the last row and the last column can contain non-zero entries. 

*The matrix entries can take values only from $\{0,1\}$. It is a kind of binary matrix. 
I am interested in the eigenvalues of this matrix. What can we say about them? In particular, when are all of them positive? 

At first I thought I understood your question, but after reading those comments and answers here, it seems that people here have very different interpretations. So I'm not sure if I understand it correctly now.
To my understanding, you want to find the eigenvalues of
$$
A=\begin{pmatrix}0_{(n-1)\times(n-1)}&u\\ v^T&a\end{pmatrix},
$$
where $a$ is a scalar, $u,v$ are vectors and all entries in $a,u,v$ are either $0$ or $1$. The rank of this matrix is at most $2$. So, when $n\ge3$, $A$ must have some zero eigenvalues. In general, the eignevalues of $A$ include (at least) $(n-2)$ zeros and
$$\frac{a \pm \sqrt{a^2 + 4v^Tu}}{2}.$$
Since $u,v$ are $0-1$ vectors, $A$ has exactly one positive eigenvalue and one negative eigenvalue if $v^Tu>0$, and the eigenvalues of $A$ are $\{a,0,0,\ldots,0\}$ if $v^Tu=0$.
