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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.

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Think of it as a game. The pieces are the entries of your matrix. The moves are the elementary row operations. You win when you get to the identity matrix. So... what's your strategy?

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 $0$.

In this fashion, moving left-to-right, we systematically clear the columns of the matrix and when we're done, we have the identity matrix. If we can't get the identity matrix this way, then we've proven that the matrix is not invertible!

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