Prove that if each row of a matrix sums to zero, then it has no inverse.

Hint: You can sum the elements of a row by multiplying this row with a vector of $1$'s. Can you find now a matrix $X$ (with appropriate columns) such that $AX=Ο$?

If the sum of the rows is zero, then the matrix has the eigenvalue $0$. As a result its $\ker$ is of dimension $\ge1$, i.e. there is a nonzero solution to $AX=0$, hence it's noninvertible

Assuming $A$ is an $n \times n$ matrix, let $v_0 = (0,0,\dots,0)$ and $v_1 = (1,1,\dots,1)$ be $n$-element column vectors. Since each row of $A$ sums to zero, it follows that $$A v_1 = (0,0,\dots,0) = A v_0,$$ showing that $A$ cannot have a (left) inverse.