# Why is that if every row of a matrix sums to 1, then the rows of the inverse matrix sums to 1 too?

Why is that if every row of a matrix sums to $1$ then the rows of its inverse matrix sum to $1$ too?

For example, consider

then its inverse is

which satisfies the condition. Is it true for every such matrix?

Let $v = (1, 1, \ldots , 1)'$ be a column vector of all $1$s. Then the rows of $A$ adding to $1$ is equivalent to saying $Av = v$. So when $A$ is invertible, we will have Thus $A^{-1}$ has rows summing to $1$ as well. (Note that $A$ will not always be invertible.)