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For the Identity matrix $(\mathbf{I_4})$,

Applying the row operation(s):

$2\cdot R_1 - R_3 \rightarrow R_3 $,

Would give the following matrix:

$$\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 2&0&-1&0 \\ 0&0&0&1 \\ \end{pmatrix} $$.

But this matrix isn't an elementary matrix, as there are essentially 2 row operations being performed right?

If not, what would the row operation be, to find the inverse of the elementary matrix?

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We negated row three, then added twice row one. These are two separate elementary row operations. The matrix you've given is the product of the "negate row three" elementary matrix (we'll denote this $A$), and the "add twice row 1 to row 3" elementary matrix (which we'll denote $B$); that is, $BA$ is the matrix you've listed.

You should know how to find the inverse of $A$ and $B$ by themselves, since they're elementary. To find $(BA)^{-1}$, use the rule that $(BA)^{-1} = A^{-1}B^{-1}$.

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