Let $A_{nxn}$ be an invertible matrix such as that the sum of the elements of each row is equal to $c$.
Prove that the sum of the elements of every row in $A^{-1}$ is equal to
$d$ and write $d$ using $c$.
Clue: you can use the column vector $v=(1,1,1,…,1)^t$.
This is the question I got and I have no idea how to solve it.
Best Answer
Start with $$A\begin{bmatrix}1\\1\\...\\1\end{bmatrix}=\begin{bmatrix}c\\c\\...\\c\end{bmatrix}$$
This is true by assumption. Now A is invertible, so multiply with the inverse from the left and see what happens:
$$\underbrace{A^{-1}A}_{I}\begin{bmatrix}1\\1\\...\\1\end{bmatrix}=A^{-1}\begin{bmatrix}c\\c\\...\\c\end{bmatrix}$$
So $$\begin{bmatrix}1\\1\\...\\1\end{bmatrix}=A^{-1}\begin{bmatrix}c\\c\\...\\c\end{bmatrix}$$
What does that tell you?