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

Just look at what needs to be done.
First, eliminate the $-5$ using $E_1 = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}$. This gives
$$ E_1 A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}.$$
Can you figure out what $E_2$ must be? 
