I'd put in a brief review of row reduction operations here and what row reduced form is, but I think the reader would be better served by reviewing the material. I dislike describing math without full rigor that describes how to handle exceptional cases, but in doing so I'd just be repeating definitions from a book. We largely just want a matrix where each row is of the form [0,0,0,1,0,0,0,*,*,*] - where we have zeros for every entry where a different row has a leading one.
The row reduction operations we want will be 1.) divide third row by 2 to get [0,0,1,3] in row 3. 2.) Subtract row 3 from row 2 to get [0,1,0,1] in row 2 3.) Add 3 times row 3 to row 1 to get [1,0,0, 12] in row 1. This will get the matrix [[1,0,0,12],[0,1,0,1],[0,0,1,3]].
The elementary matrix for step 1 is E_1=[[1,0,0],[0,1,0],[0,0,1/2]], and the matrix we get from the operation is E_1 * A. The elementary matrix for step 2 is E_2=[[1,0,0],[0,1,-1],[0,0,1]], and the matrix we get from the operation is E_2 * E_1 * A. The elementary matrix for step 3 is E_3 = [[1,0,3],[0,1,0],[0,0,1]], and the matrix we get is E_3 * E_2 * E_1.
Therefore E=E_3 * E_2 * E_1 = [[1,0,3/2],[0,1,-1/2],[0,0,1/2]] and E*A = [[1,0,0,12],[0,1,0,1],[0,0,1,3]].
