Finding Inverse of a matrix using elementary transformations

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So I have to find the Inverse of A.

$$ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \\ \end{bmatrix} $$ By using elementary row or column transformations..

The method involves writing:

$A A¯¹ = I $

And then performing the same set of operations on both A and I, to convert A to I and then we get the Inverse on the R. H. S of the equation.

So by a series of transformations I get $$ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 4 \\ \end{bmatrix} $$

It is wrong but my steps seemed to be legit but it took too many operations thus mistake is possible.

So can someone suggest me some steps so that A converts to I in minimum transformations.

I = Identity Matrix