## 每週問題 April 22, 2013

Let $A$ be an invertible matrix. If the sum of the elements of every row of $A$ is equal to $k$, show that the sum of the elements of every row of $A^{-1}$ is equal to $1/k$.

$\displaystyle A^{-1}\mathbf{e}=A^{-1}\left(\frac{1}{k}A\mathbf{e}\right)=\frac{1}{k}A^{-1}A\mathbf{e}=\frac{1}{k}\mathbf{e}$

PowSol-April-22-13