Prove that if each row of a matrix sums to zero, then it has no inverse. Could anyone help me with this proof without using determinant? I tried two ways. 

Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix. 

I then get stuck. The other way was to prove by contradiction, and I failed too. 

If each row of $A$ sums to zero, then each row of the column vector that is the sum of the column vectors constituting $A$ is zero. So the columns of $A$ are not linearly independent, and therefore the matrix is singular (i.e. it has no inverse).
