# what is the easiest way to find the inverse of a 3x3 matrix by elementary column transformation?

While using the elementary transformation method to find the inverse of a matrix, our goal is to convert the given matrix into an identity matrix.

We can use three transformations:- 1) Multiplying a column by a constant 2) Adding a multiple of another column 3) Swapping two column

The thing is, I can't seem to figure out what to do to achieve that identity matrix. There are so many steps which I can start off with, but how do I know which one to do? I think of one step to get a certain position to a 11 or a 00, and then get a new matrix. Now again there are so many options, it's boggling.

Is there some specific procedure to be followed? Like, first convert the first column into: 1 a12 a13
0 a22 a23 0 a32 a33

Then do the second column and then the third?

What do I start off with? I hope I've made my question clear enough.

The strategy I prefer goes like this. We want a $1$ in the upper left corner and $0$s above and below it. So let's use row operations to make sure the upper left corner has a nonzero entry. Now let's use that entry to make all the entries below it $0$.
At this point, the leftmost column is exactly what we want it to be! Now we move to the second column. We want the second entry of that column to be $1$, so put a nonzero entry there using row operations, and then use row operations to make all entries above and below it $0$.