I understand when using Gaussian Elimination you have to get it in ref form (upper triangle) and calculate the product of the diagonal. Additionally you have to keep track of the number of swaps to determine how many times to multiply by negative one.
Let us say you have the following matrix: $\begin{pmatrix} 2&-1\\ -1&-1\\ \end{pmatrix}$
and you scale the top row by 1/2 making $\begin{pmatrix} 1&-1/2\\ -1&-1\\ \end{pmatrix}$ <-this step is not needed, but let us say we do it
then you do do P1+P2 into P2 giving $\begin{pmatrix} 1&-1/2\\ 0&-3/2\\ \end{pmatrix}$
so the determinant is -3/2 according to this, but the actual determinant is -3 (you can get that by skipping my unnecessary 2nd step as it is not needed to get the answer in REF.
My question is why does scaling the top affect the determinant, I thought the only things that you had to look out for where row swaps to determine the amount of times you multiply by negative 1.
$\DeclareMathOperator{Row}{Row}$Gaussian elimination can be rigorously studied by way of elementary matrices. In this case, we wish to compute $\det(A)$ where $$ A= \begin{pmatrix} 2 &-1\\-1&-1 \end{pmatrix} $$ The plan is to row-reduce $A$, keeping track of our steps with elementary matrices.
Step 1. Multiply $\Row_1$ by $1/2$. This corresponds to multiplying $A$ on the left by $$ E_1=\begin{pmatrix}1/2 & 0 \\ 0 &1\end{pmatrix} $$ and the result is $$ E_1 A=\begin{pmatrix}1 & -1/2\\ -1 & -1\end{pmatrix} $$
Step 2. Add $\Row_1$ to $\Row_2$. This corresponds to multiplying $E_1 A$ on the left by $$ E_2=\begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix} $$ and the result is $$ E_2 E_1 A=\begin{pmatrix}1 & -1/2\\ 0 & -3/2\end{pmatrix} $$
Step 3. Multiply $\Row_2$ by $-2/3$. This corresponds to multiplying $E_2E_1 A$ on the left by $$ E_3=\begin{pmatrix}1 &0\\ 0 & -2/3\end{pmatrix} $$ and the result is $$ E_3E_2E_1A=\begin{pmatrix}1 & -1/2\\ 0 & 1\end{pmatrix} $$
Now, we have $$ \det(E_3)\det(E_2)\det(E_1)\det(A)=1 $$ Moreover, it is easy to compute $\det(E_1)$, $\det(E_2)$, and $\det(E_3)$. Can you compute these determinants and use them to find $\det(A)$?