Solving problem with two invertible matrices using Gauss elimination method

Question

So I solved this problem located in the picture below but I am not very confident I did it right so can someone look at it and see if it was solved right? Thank you! This phrase "Remark/Warning: Computing the product Ab^(-1) A is very bad idea" has me a little concerned about if I am correct or not.

Problem

Answer 1

$AB^{-1}A\vec{x} = \vec{b}$

$A^{-1}B^{-1}A\vec{x} = A^{-1}\vec{b}$

$(BA)^{-1}A\vec{x} = A^{-1}\vec{b}$

$ (BA)(BA)^{-1}A\vec{x} = BAA^{-1}\vec{b}$

$ IA\vec{x}= BI\vec{b}$

$ A\vec{x} = B\vec{b}$

$A^{-1}A\vec{x} = A^{-1}B\vec{b}$

$I \vec{x} = A^{-1}B\vec{b}$

$\vec{x} = A^{-1}B\vec{b}.$