Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 23/Hint 2

Instead of doing this for each column, we can row reduce all these systems simultaneously, by attaching all columns of $I$ (i.e. the whole matrix $I$) on the right of $A$ in the augmented matrix and obtaining all columns of $X$ (i.e. the whole inverse matrix) on the right of the identity matrix in the row-equivalent matrix:

$[A|I]\to [I|X]$.

If this procedure works out, i.e. if we are able to convert $A$ to identity using row operations, then $A$is invertible and $A^{-1}=X$.