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I want to make an upper triangular matrix. From this:

$\begin{bmatrix}2&0&1\\0&1&1\\3&1&0 \end{bmatrix}$

The first is the correct one. The second is incorrect, yet I fail to understand why.

$\begin{bmatrix}2&0&1\\0&1&1\\3&1&0 \end{bmatrix}$

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

$\begin{bmatrix}2&0&1\\0&1&1\\0&0&-5/2 \end{bmatrix}$

Second method.

$\begin{bmatrix}2&0&1\\0&1&1\\3&1&0 \end{bmatrix}$

$\begin{bmatrix}1&0&1/2\\0&1&1\\3&1&0 \end{bmatrix}$ As you can see, I've immediately divided the first row by 2 so that I can get a pivot of 1.

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

$\begin{bmatrix}1&0&1/2\\0&1&1\\0&0&-5/2 \end{bmatrix}$

What happened? Is dividing a row by an integer illegal?

[edit.] The reason for the upper triangular matrix, is because I want to find the determinant.

I am aware that switching rows causes the sign to change. No big issue, because I can note this, and at the end adjust the determinant accordingly. But is this similar? Is the returned determinant supposed to be re-multiplied because I had divided it first?

$\begin{bmatrix}2&0&1\\0&1&1\\3&1&0 \end{bmatrix}$

The first is the correct one. The second is incorrect, yet I fail to understand why.

$\begin{bmatrix}2&0&1\\0&1&1\\3&1&0 \end{bmatrix}$

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

$\begin{bmatrix}2&0&1\\0&1&1\\0&0&-5/2 \end{bmatrix}$

Second method.

$\begin{bmatrix}2&0&1\\0&1&1\\3&1&0 \end{bmatrix}$

$\begin{bmatrix}1&0&1/2\\0&1&1\\3&1&0 \end{bmatrix}$ As you can see, I've immediately divided the first row by 2 so that I can get a pivot of 1.

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

$\begin{bmatrix}1&0&1/2\\0&1&1\\0&0&-5/2 \end{bmatrix}$

What happened? Is dividing a row by an integer illegal?

[edit.] The reason for the upper triangular matrix, is because I want to find the determinant.

I am aware that switching rows causes the sign to change. No big issue, because I can note this, and at the end adjust the determinant accordingly. But is this similar? Is the returned determinant supposed to be re-multiplied because I had divided it first?

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