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? 

Such a matrix, of size $3 \times 3$ or larger, will never have all non-zero eigenvalues.  This is because the first two columns of the matrix are linearly independent, hence the determinant is zero, hence zero is an eigenvalue.
