We were taught in linear algebra that in order to try to find the inverse of a matrix we can create an augmented matrix $[AI]$ where $A$ is the original matrix and $I$ is the identity matrix. Then we just row reduce everything and if it all checks out, the right hand side of $[AI]$ will be $A^{{-1}}$.
However, there is a brief mention in the textbook about an alternative perspective on this without providing further explanation, proof, or examples.
This is my best attempt at an interpretation of what it is trying to convey:
$A^{{-1}}$ is the matrix composed of columns $[x_1 x_2 x_n]$ where $x_1, x_2, x_n$ are solutions to the equations $Ax_1=e_1,Ax_2=e_2,Ax_n=e_n$. Where $e_n$ are the respective columns of the identity matrix
Is my comprehension of the following correct? Please see original text below:
