[Math] Inverse of matrix whose row sums is 1

linear algebramatrices

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}

Best Answer

"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.

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