Inverse of $\{a_1 A_1,...,a_n A_n\}$

Question

$a_1,...,a_n\in \mathbb{R}$

$A_1,...,A_n$ are the rows of the invertible matrix A

I am trying to find a regular formula for this. Is it possible?

Thanks for help!

Answer 1

Let $A=\{ A_1,\ldots, A_n\}$, $B=\{a_1 A_1,\ldots,a_n A_n\}$. Then $B=AC$ where

$$C=\left( \begin{array}{cccc} a_1 & 0 & \ldots & 0 \\ 0 & a_2 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots \\ \ldots & \ldots &\ldots  & \ldots \\ \end{array} \right)$$ Hence $B^{-1}=C^{-1}A^{-1}$