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

A=(1/32/33/41/4)

then its inverse is

A−1=(−3/58/59/5−4/5),

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

**Answer**

Let v=(1,1,…,1)′ be a column vector of all 1s. Then the rows of A adding to 1 is equivalent to saying Av=v. So when A is invertible, we will have A−1v=A−1Av=v Thus A−1 has rows summing to 1 as well. (Note that A will not always be invertible.)

**Attribution***Source : Link , Question Author : Garmekain , Answer Author : Suhas Vijaykumar*