Inverse of matrix whose row sums is 1 Given a $10 $x$ 10$ invertible matrix with real entries, the sum of whose rows is $1$, prove that for the inverse, the sum of the elements in each row is $1$ ?  {Question from TIFR GS-2015}
"Sum of all rows equals $1$" means precisely that the vector $(1, \dotsc, 1)^T$ is a fixed point. Of course this vector is then a fixed point of the inverse, too.
Let $\textbf{1}$ be $10\times1$ vector with each component $1$ . Then $A(\textbf{1})=1(\textbf{1})$. So, $1$ is an eigenvalue of A, thus of $A^{-1}$ also. Thus, $1(\textbf{1})=A^{-1}(\textbf{1})$ (left multiply by $A^{-1}$), which implies that the sum of each row of the inverse matrix is also $1$.