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. 

Hint: For a matrix $A$ having such a property has vector $(1,1,...,1)$ in its kernel thereby giving $\dim(\ker A)>0$.
