Find the inverse of the matrix using elementary matrices. [[2,0],[1,1]]

Let the given matrix be,
$A=[[2,0],[1,1]]$
Now transform the given matrix in to unit matrix I by performing column reduction to get,
$A=[[2,0],[1,1]]$
$=[[2,0],[1,1]](c_1->c_1-c_2)$
$=[[1,0],[0,1]](c_1->1/2\ast c_1)$
=I
Now, construct sequence of elementary matrices such that E2*E1 A=I
The column operation are,
$E_1=[[1,0],[0,1]](c_1->c_1-c_2)$
$=[[1,0],[-1,1]]$
$E_2=[[1,0],[0,1]](c_1->1/2\ast c_1)$
$=[[1/2,0],[0,1]]$
Then the inverse of the matrix by elementary matrices is given by:
$A^{-1}=E_1\times E_2$
$=[[1,0],[-1,1]]\ast [[1/2,0],[0,1]]$
$=[[1/2,0],[-1/2,1]]$