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

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

then its inverse is

A1=(3/58/59/54/5),

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 A1v=A1Av=v Thus A1 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

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