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I'm really confused with the Jordan Exchange (or pivot operation) for non pivot rows and columns. I apologize if this seems to be easily googled, but I've been struggling and I'm not sure what I'm doing wrong given the formulas.

For instance, given the matrix

\begin{bmatrix}1&4&3&-2\\-1&2&1&0\\ -4&2&0&2\end{bmatrix}

I understand that for the first row, we simply replace the numbers with the row values divided by the negative inverse. So, assuming our pivot is (1,1), our first row would be:

\begin{bmatrix} 1&-4&-3&2 \\ \\ & \end{bmatrix}

And for the pivot row, we simply replace it the values divided by the pivot:

\begin{bmatrix} 1&-4&-3&2 \\ -1& & & \\ -4& & & \end{bmatrix}

But now I'm confused with the remaining pivots. I understand the general formula is $A_{ij} - \frac{A_{is}A_{rj}}{A_{rs}}$, and I'm getting the correct values for certain things. For instance, in R2C2 (2), I understand I obtain its value after pivoting by using $2 + \frac{(4)(-1)}{1} = 6$, but for R3C2 I'm not really sure what to do.

Similarly, I'm not getting correct values for R2C3 (1). I'm calculating it via $1-\frac{3*2}{1} = -5$, but the actual value should be 4.

The final value should be:

\begin{bmatrix} 1&-4&-3&2 \\ -1&6&4&-2 \\ -4&18&12&-6 \end{bmatrix}

Help would be much appreciated!

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You always have a kind of rectangle. We update the element $a_{23}=\color{blue}1$, where $a_{11}=\color{red}1$ is the pivot element.

$$\begin{bmatrix}\color{red}1&4&\color{magenta}3&-2 \\ \color{magenta}{-1}&2&\color{blue}1&0\\ -4&2&0&2\end{bmatrix}$$

Thus $a_{23}^*=a_{23}- \large{\frac{a_{21}\cdot a_{13}}{a_{11}}}=\color{blue}1-\frac{\color{magenta}{(-1)}\cdot \color{magenta}3 }{\color{red}1}\normalsize=1+3=4$


Similar calculation for $a_{34}=2$

$$\begin{bmatrix}\color{red}1&4&3&\color{magenta}{-2} \\ {-1}&2&1&0\\ \color{magenta}{-4}&2&0&\color{blue}2\end{bmatrix}$$

$a_{34}^*=a_{34}- \large{\frac{a_{31}\cdot a_{14}}{a_{11}}}\normalsize=2-\frac{(-4)\cdot (-2)}{1}=-6$

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