Solve 2x2 Matrix in 30 Mins: An Attempt

[nicknaq]

I really need an answer in the next 30 min.

Homework Statement

Given the 2x2 matrix A= 0 −2
9 −5
Write A^-1 as a product of 4 elementary matrices.

The Attempt at a Solution

I got the following. It's wrong. Please find my error.

1 0 | 1/9 0 | 1 0 | 1 5/9
0 1 | 0 1 | 0 -1/2 | 0 1

Edit: Geeze. the forum won't let me show the spaces between the four matrices above. sorry.

 

[HallsofIvy]

After swapping the two rows, dividing the first row by 9, and dividing the second row by -2, you have row-reduced A to
$$\begin{bmatrix}1 & -5 \\ 0 & 1\end{bmatrix}$$
so you have to add 5 times the second row to the first row. What elementary matrix does that?
(You seem to have kept the "9" you got rid of with the second elementary matrix.)