Elementary Matrix Multiplication and Gauss Elimination

Question

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 $9 + (-3)\times 1 = 6 \neq 5$

Answer 1

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 $A^2$.