Recall, from the hint, that row operations can be used to write the inverse of a matrix. We set up the augmented matrix [ A | I ] {\textstyle [A\ |\ I]} , once transformed using row operations so that the identity is on the left, it necessarily shows A − 1 {\textstyle A^{-1}} on the right, [ I | A − 1 ] {\textstyle [I\ |\ A^{-1}]} .
We conclude