Find Elementary Matrics E1 and E2 such that $E_2E_1$A = I I am studying Linear Algebra part-time and would like to know if anyone has advice on solving the following type of questions:
Considering the matrix:
$$A = \begin{bmatrix}1 & 0 & \\-5 & 2\end{bmatrix}$$
Find elementary Matrices $E_1$ and $E_2$ such that $E_2E_1A = I$
Firstly can this be re-written as?
$$E_2E_1 = IA^{-1}$$
and that is the same as?
$$E_2E_1 = A^{-1}$$
So I tried to find $E_1$ and $E_2$ such that $E_2E_1 = A^{-1}$:
My solution:
$$A^{-1} = \begin{bmatrix}1 & 0 & \\{\frac {5}{2}} & {\frac {1}{2}}\end{bmatrix}$$
$$E_2 = \begin{bmatrix}1 & 0 & \\0 & {\frac {5}{2}}\end{bmatrix}$$
$$E_1 = \begin{bmatrix}1 & 0 & \\1 & {\frac {1}{5}}\end{bmatrix}$$
This is the incorrect answer. Any help as to what I did wrong as well as suggestions on how to approach these questions would be aprpeciated.
Thanks

Thanks for the advice and hints TenaliRaman and copper.hat. I'll post the solution here for anyone else who needs to solve something similar.
So we have to find the elementary matrices $E_1$ and $E_2$ such that $E_2E_1A = I$:
Start by eliminating the -5 in matrix A.
So find $E_1$ so that 
$$E_1A = \begin{bmatrix}1 & 0 & \\0 & 2\end{bmatrix}$$
the matrix $E_1$ is thus:
$$E_1 = \begin{bmatrix}1 & 0 & \\5 & 1\end{bmatrix}$$
If we rewrite the original equation like so: 
$$E_2(E_1A) = I$$
Find $E_2$ (since we know what $E_1A$ is)
Solving this we get:
$$E_2 = \begin{bmatrix}1 & 0 & \\0 &  {\frac {1}{2}}\end{bmatrix}$$
Thanks everyone!
