Yes, that's right. A "row operation" is one of three kinds:

1) swap two rows

2) multiply every member of a row by the same constant.

3) replace every member of a row by itself plus a constant time the corresponding member of a second row.

An "elementary matrix" is a matrix constructed from the identity matrix by a single row operation.

What you want to do is reduce the given matrix to the identity matrix (which is possible because it is invertible) by row operations,writingdown the elementary matrix corresponding to each row operation.

For example, starting from

My first steps would be to reduce the first column to the column .

The number in the upper left is already 1 so I don't need to change that. I can get 0 in the next row by subtracting the first row from the second row. The elementary matrix corresponding to that is

To get 0 in the third row, I need to subtract the first row from the third row. That gives the elementary matrix

and they convert the matrix to

Now you need to convert the second column properly and then the third. Can you continue?

After you have converted the matrix to the identity matrix and written down all the corresponding elementary matrices (in the proper order) so they multiply to give the original matrix.

Normally it would take 9 row operations to reduce a 3 by 3 matrix to the identity matrix and so the matrix would be represented as the product of 9 elementary matrices. Notice that the upper left number wasalready1 so we didn't need a row operation for that. Also that 0 now in the second row, third column will stay there we won't need a row operation to change that. I suspect that sort of thing, a 0 or 1 already in the correct position will happen once more to give 6 rather than 9 elementary matrices.