<s>[INST] Elementary Matrix Multiplication and Gauss Elimination I have the following matrix:
$$ A = 
\begin{bmatrix}
    2 &  1 &  1 &  0 \\
    4 &  3 &  2 &  1 \\
    8 &  7 &  8 &  5 \\
    6 &  7 &  9 &  8 \\
\end{bmatrix}
$$
and I want to apply Gauss elemination through a series of elementary matrix multiplications. 
To cancel out the numbers under the first diagonal element of $A$, $$E_1 = 
\begin{bmatrix}
    1 &  0 &  0 &  0 \\
    -2 &  1 &  0 &  0 \\
    -4 &  0 &  1 &  0 \\
    -3 &  0 &  0 &  1 \\
\end{bmatrix}
$$
Applying $E_1$, it works as expected 
$$A^1 = E_1A = \begin{bmatrix}
    2 &  1 &  1 &  0 \\
    0 &  1 &  0 &  1 \\
    0 &  3 &  4 &  5 \\
    0 &  4 &  6 &  8 \\
\end{bmatrix}$$
For the second column, however, starting with $E_2 = \begin{bmatrix}
    1 &  0 &  0 &  0 \\
    0 &  1 &  0 &  0 \\
    0 &  0 &  1 &  0 \\
    0 &  0 &  0 &  1 \\
\end{bmatrix}$, how do I eliminate numbers under the second diagonal element of $A$? 
If I want to apply on $E_2$ the row operations that would make the numbers under the second diagonal element of $A$ zero, then I would get for the third row of $E_2$, assuming it starts as an identity matrix, $I_n$ wouldn't I get:
$$E_2 = \begin{bmatrix}
    1 &  0 &  0 &  0 \\
    0 &  1 &  0 &  0 \\
    -7 &  0 &  1 &  0 \\
    -7 &  0 &  0 &  1 \\
\end{bmatrix}$$
The row operations in the context of A are apply $-7R_1 + R_3$ and $-7R_1 + R_4$ to cancel out the numbers under the second diagonal element of $A$ - so, the above shows I applied those to $I_n$ 
Doing the multiplication though, I get 
$$A^2 = E_2E_1A = \begin{bmatrix}
    2 &  1 &  1 &  0 \\
    0 &  1 &  0 &  1 \\
    -14 &  -4 &  -3 &  5 \\
    -14 &  -3 &  -2 &  8 \\
\end{bmatrix}$$
What am I doing wrong?
Edit: Calculation error in $A^1$ at position $(4,3)$. It should be $9 + (-3)\times 1 = 6 \neq 5$ 
 [/INST] First of all, you are applying $E_2$ to $A^1$. Hence, while designing $E_2$, you should be looking at $A^1$ (and not at $A$). While looking at $A_1$, you know that you want to cancel $3$ and $4$ at position $(3,2)$ and $(4,2)$ respectively (and not $-7$ of $A$).
You can try $E_2$ as follows:
$$E_2 = \left[
\begin{array}{cccc}
1& 0 & 0&0\\
0&1&0&0\\
0&-3&1&0\\
0&-4&0&1
\end{array} \right] $$
The reason to select this  $E_2$ is that you want to operate $-3R_2 + R_3$ at $R_3$ and $-4R_2 + R_4$ at $R_4$. Hence, the third row of $E_2$ should have $-3$ and $+1$ in the second and third column respectively. Similarly, the fourth row of $E_2$ should have $-4$ and $+1$ in the second and fourth column.
In addition to the reason mentioned in the beginning, your $E_2$ also doesn't work because $-7$ in your $E_2$ operates on first column. If you want to limit the operation on the second column, the first column of $E_2$ should remain $[1\ 0\ 0\ 0]$. 
At the end, you should get 
$$A^2 = E_2 E_1 A = \left[
\begin{array}{cccc}
2& 1 & 1&0\\
0&1&0&1\\
0&0&4&2\\
0&0&6&4
\end{array} \right] $$
Now, you can design $E_3$ by using the operation $-\frac{3}{2}R_3 + R_4$ at $R_4$ of $A^2$.
</s>